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The pion-nucleon interaction at low energy Bagheri, Ardeshir 1986

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THE PION-NUCLEON INTERACTION AT LOW ENERGY by ARDESH1R BAGHERI B . S c , P a h l a v i U n i v e r s i t y , 1978 M.Sc., Indiana U n i v e r s i t y , 1980 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Phys i c s We accept t h i s t h e s i s as conforming to the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y 1986 ° A r d e s h i r Bagheri, 1986 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date o f . t 14. I 5 ( 3 / 8 1 ) ( i i ) ABSTRACT The ir~p * yn and n~p • ir°n reactions were studied for laboratory pion kinetic energies of 50, 66, 80, 95, 110, and 125 MeV at 9 angles between 30° and 140°, using a large Nal(Tl) detector, TINA. The results are much more accurate than previous data. The radiative capture differential cross-sections are compared with previous data and with several calculations and w i l l specify more precisely the isoscalar amplitudes i n multipole analyses. They are consistent with earlier data and confirm the principle of detailed balance, but indicate that the theoretical calculations cannot reproduce the nucleon data adequately below resonance. The charge exchange differential cross-sections are obtained by unfolding the energy spectra of the TT° y-ray decay, and agree with the Karlsruhe and VPI phase-shift analyses. Pion-nucleon phase-shifts and scattering lengths are calculated and the results are in good agreement with data from the elastic scattering channels and support isospin invariance in this energy region. ( i i i ) Table of Contents Abstract Table of Contents L i s t of Tables L i s t of Figures Acknowledgements Chapter I Introduction 1.1.1 General introduction 1.1.2 Low energy pion-nucleon scattering 1.2 Technical advantages 1.3 Charge exchange reaction and hadronic isospin symmetry 1.4 Radiative capture and general study of photomeson reactions 1.5 A review of previous work Chapter II The Experiment Characteristics of low energy pion beams The pion beam line at TRIUMF The liquid hydrogen target The Nal(Tl) detector Characteristics of alka l i halide sc i n t i l l a t o r s The TINA detector The Experiment Set-up and event definition Plastic s c i n t i l l a t i o n counters Electronics The test run Chapter III Data Acquisition and Analysis Data acquisition Data analysis Introduction The response function Cuts and the production of f i n a l spectra Radiative capture data Analysis of the radiative capture part of the spectra Application of the principal of detailed balance to pi-meson photoproduction The analysis of the charge exchange part of the spectra The renormalization of the 50 and 60 MeV data II.1 II.2 II.3 II.4 II.4.1 II.4.2 II.5 II.5.1 II.5.2 II.5.3 II.5.4 I I I . l III.2 III.2.1 III.2.2 III.2.3 III.3 III.3.1 III.3.2 III.4 III.5 (iv) Chapter IV The Theory of Pion Interactions with Nucleons IV .1 Introduction IV .2 On energy-shell amplitudes; partial waves IV.3.1 Hadronic Isospin symmetry IV .3 .2 Isospin and the pion-nucleon system IV .4 .1 Phase shift analysis IV .4 .2 Determination of phase shifts IV .5 Optical theorem and total cross section IV .6 Pion-nucleon dynamics, Born terms IV .7 The theory of pion photoproduction IV .7 .1 The h e l i c i t y , multipole amplitudes, and multipole analysis IV .7 .2 The electromagnetic interaction and isospin IV .7 .3 The isotensor term of the electromagnetic current IV .7 .4 Time reversal invariance Chapter V Experimental Results and Discussions V .1 Introduction V .2 .1 Radiative capture results V.2.2 The photoproduction data V .2 .3 The radiative capture data V .3 .1 Comparison with multipole analysis V .3 .2 Comparison with other theoretical calculations V .4 Time reversal invariance V .5 .1 The charge exchange results V.5.2 Comparison with phase shift analysis and other data V .5 .3 Pion-nucleus interaction and free pion-nuclear excitation function V.5.4 Conclusion L i s t of References (v) L i s t of Tables TABLE II.1 TABLE II.2 TABLE II.3 TABLE II.4 TABLE I I I . l TABLE IV.1 TABLE IV.2 TABLE IV.3 TABLE IV.4 TABLE IV.5 TABLE IV.6 TABLE IV .7 TABLE V.l TABLE V.2 TABLE V.3 TABLE V.4 TABLE V.5 TABLE V.6 Kinematics of the Reactions (it~p ••• yn) and (it~p •> Ti 0n) Pion Beam Fraction at Different Energies Pion Beam Energy Spread Dimensions and Efficiencies of Plastic Scintillation Counters Results of Monte Carlo Stimulation for Response Function Parameters C and D The it-Nucleon State Functions |nN> in Terms of Isospin State Function | l , I 3> Isospin Scattering-Amplitude Coefficients for it-N Reactions Pion-Nucleon Scattering Length and Volume Lowest Energy Resonances in the Pion-Nucleon System The Matrix Elements f-J^ The Contribution of the Multipoles EL and ML to Photoproduction Multipole Analysis of Single Pion Production Experimental Results for the Reaction (Tt -p -• yn) Experimental Results for the (TI~P -*• u°n) Reaction Phase Shifts (in degrees) Obtained from the Data after Applying the Electromagnetic Corrections The S-wave Scattering Length Obtained from This Experiment and Other References List of the Pion-Nucleon Scattering Experiments Used in Phase-Shift Analyses Differential Cross Sections for the (it~p •* ir°n) Reaction at 0° and 180° PAGE PAGE PAGE PAGE PAGE PAGE PAGE PAGE PAGE 32 35 36 PAGE 38 PAGE 56 PAGE 90 93 104 107 110 111 115 PAGE 131 PAGE 159 PAGE 164 PAGE 165 PAGE 166 PAGE 178 (vi) L i s t of Figures Figure 1.1 Page 11 ( i r +p) and (ir~p) total cross sections. The curves are calculated from phase shifts which were fi t t e d to the world ir-nucleon data using the program SAID. Figure 1.2 Page 13 The low energy pion-nucleon relations of Brueckner, Serber and Watson. Figure II.1 Page 21 The Jacobian of transformation from the rest frame of a 60 MeV pion to the laboratory frame. Since the angular distribution of the decay muon is isotropic in the rest frame of the pion, the angular distribution of the decay muon in the laboratory would also have the same shape as the figure. Figure II.2 Page 23 The layout of Mil pion channel. Figure II.3 Page 26 Energy band structure of an activated crystalline s c i n t i l l a t o r . Figure II.4 Page 31 Experimental set-up. SI, S2, S3 and S4 are s c i n t i l l a t i o n counters. TINA i s a Nal(TJl) Y~ray detector, movable around the target from 0° to 145°, and is surrounded by an iron shield 25 cm thick at the front and 10 cm thick on the sides. Figure II.5 Page 34 The time distribution of the beam particles (S1.S2.S3) with respect to the RF buckets of the primary proton beam (T^ = 66 MeV). The beam composition at S3 is obtained from this histogram. Figure II.6 Page 39 Schematic diagram of electronics. Figure II.7 Page 42 The time distribution of neutral events in the NaI(T£) crystal, obtained by a "start" signal from S1.S2.S3 coincidence and stop signal from the crystal (T^ = 62 MeV). ( v i i ) Figure II.8 Page 44 Results of the test run for different ADCs using an e + beam. Figure II.9 Page 45 Results of the test run for different ADCs using stopped pion gamma ray. Figure I I I . l Page 49 The "large detector" extreme in gamma ray spectroscopy. A l l gamma ray photons, no matter how complex their mode of interaction, ultimately deposit a l l their energy in the detector. Some representative histories are shown at the top (KN079). Figure III.2 Page 53 A plot of the response function of the NAI(Tfc) detector. Figure III.3 Page 54 The stopped pion gamma ray spectrum in TINA using a 25.4 cm collimator. Figure III.4 Page 55 The stopped pion gamma ray spectrum in TINA using a 15.24 cm collimator. Figure III.5 Page 59 The time distribution of beam particles (T u • 110 MeV). Figure III.6 Page 60 The time distribution of neutral events in the TINA crystal (T^ " 110 MeV). This histogram was used to separate gammas from neutrons by accepting only events within the window indicated. Figure III.7 Page 62 Typical y-ray energy spectra in TINA with a l l cuts applied and the f i t . Figure III.8 Page 66 The y-ray spectra and the f i t for (ir~p •»• yn) reaction at several different energies and angles. The background for the ir° decay ( T T ° •*• 2y) is also shown. ( v i i i ) Figure III.9 Page 77 The charge exchange y-ray spectra and the f i t at several different energies and angles. The contribution of different Legendre polynomial terms are also shown. Figure IV.1 Page 101 Diagram for Born terms: (a) direct; (b) crossed. Figure IV.2 Page 106 Born terms for single pion production. Figure IV.3 Page 106 Isobar and heavy meson terms for single pion production. Figure V.l - V.6 Page 125 The differential cross-section for the (n -p •*• y°) reaction shown as the time reversed reaction (yn + ir~p) at six different energies. Also shown are the calculations by (SMI82), (BLA80), and (ARA85). [B & L(l)] i s the result of a multipole analysis while [B & L (2)] shows the result of a pseudoscalar-relativistic calculation. (S & Z) and (ARAI) are two other multipole analyses. Figure V.7 Page 134 The total cross-section of the (yn •»• n -p) reaction as a function of energy including previous experimental results and the theoretical results of Blomqvist and Laget (BL077). Figure V.8 Page 137 The comparison between the (yn -»• i r -p) reaction cross-section (shaded area) extracted from (yd •*• ppir") yield when P r s 50 MeV/c (ARG78), and the bubble chamber data ( f u l l circles)(ROS73), (open circles) (BEN73), and those deduced from the ( T r + / i r ~ ) ratio measurements (open triangles) (H0L74, FUJ77, FUJ72, CLI74). Figure V.9 Page 138 The comparison between Saclay data, shaded area (ARG78), and the cross-section deduced by detailed balance, from the (n~p •*• yn) cross-sections ( f u l l circles) (GUE75), (open triangles) (BER74). Figure V.10 Page 141 Preliminary results for the spectra of the pions emitted in the reactions (yd •*• nmr+) and (yd •*• ppir") at ( 9 - ) ! ^ = 45.6°, when Ey • 299 MeV. The broken line curves include only the quasi free contribution, whereas the f u l l line curves include also the f i n a l state interaction. The (yp •*• i r +n) reaction indicates the momentum resolution (LAG81). (ix) Figure V . l l Page 142 Differential cross-sections for D(Y,ir) reactions versus photon energy for 29 MeV pions at 6 i a b = 90° (FRA80, BER81). The cross hatched represents the experimental results. The line marked no FSI is the calculation without f i n a l state interaction. The curve marked B = » is for a zero range N-N interaction. The solid curve is for a f i n i t e range N-N interaction. Figure V.12 Page 144 The n~/it + ratio angular distributions when Ey - 260 MeV, Ey = 300 MeV and Ey = 350 MeV. The experimental results are from Saclay (FAU84, f u l l circles) and Tokyo (FUJ77, open c i r c l e s ) . The broken line curve is the free nucleon tCli& r a t i o . The dash-dotted line curve includes only the quasi-free amplitude. The f u l l line curves also take into account the rescattering diagrams (LAG81). Figure V.13 Page 151 The different diagrams considered in pion photoproduction on nucleons. The wavy line, broken line, and f u l l line represent respectively the photon, pion and nucleon. The A(1236) inter-mediate state Is represented by a double l i n e , (a) and (b) are the PS Born terms and PV Born terms plus S-channel A(1236) formation, (c) and (d) are time ordered decompositions of the direct and crossed nucleon Born terms (BL077). Figure V.14 Page 157 y-ray energy spectra at 27.4 MeV and 90°. One can identify clearly the (n _p •* Y n) a°d (n~p •*• it°n) events both for stopped and in - f l i g h t plons. Figure V.15 Page 160 The differential cross-sections for the charge exchange reaction at different energies, as obtained from our best f i t s . Figure V.16 Page 161 The total cross-sections for the charge exchange reaction including previous experimental results and recent phase-shift analyses. Figure V.17 Page 174 Angular distribution IAS at different energies. The curves are the result of a Legendre polynomial f i t to the data and are used to extrapolate to the cross section at zero degree (IR085). (x) Figure V.18 Page 176 The measured 0° excitation function for the free (nN) process and the IAS transition on several nuclei (IR085). Figure V.19 Page 179 The differential cross sections at 0° and 180° for (n~p -»• n°n) reaction including Saclay data (DUC73) and recent Los Alamos data (FIT85). (xi) ACKNOWLEDGEMENTS This experiment was performed by a group of people and I am grateful to each one of them. Included in this group were my supervisor, D.F. Measday, K. Aniol, M.D. Hasinoff, J .-M. Poutissou, B.C. Robertson, and M. Salomon. I wish especially to thank Martin Salomon for his invaluable help with this project. I am grateful to Farrokh Entezami for his help with the analysis, his encouragements, and above a l l for being a true friend. I am also grateful to D.S. Beder and R.M. Vfoloshyn for useful discussions. I thank Mrs. Dorothy Sample for helping me with the programming and spending several late evenings and early mornings at TRIUMF, and for her friendship and kindness. Finally, I thank Connie Zator for carefully transforming a long and complicated manuscript into an accurate typescript. I dedicate this work to my parents, Mr. and Mrs. Mohammad A l l Bagheri. - 1 -CHAPTER I  Introduction 1.1.1. General Introduction Pion nucleon scattering is one of the most fundamental subatomic inter-actions and has been studied for many years. It has played a particularly significant role in the development of particle physics. On the one hand, i t is readily accessible to experimental investigations because of the availability of high intensity pion beams over a wide range of energies. On the other hand, i t provides a testing ground for many of the new theoretical ideas. At medium and high energies, up to 2000 MeV laboratory pion energy, many phase shift analyses have been completed, and this has led to the discovery of baryon resonances. The well known (3,3) resonance A"*~'"(1232) has yielded a test of charge-independence for strong interactions, verified the formalism of f i n a l state interactions relating pion-nucleon scattering to photopion production, and stimulated the development of the Chew-Low theory. The forward scattering of pions by nucleons has served as the basis for testing the validity of microcausality through dispersion relations. Many new resonances have been found in the phase-shift analyses and these results have imposed severe constraints on the non-strange baryon quark models of the nucleon and the nN resonances. The high energy pion-nucleon scattering amplitudes, in both the forward and the backward directions, were essential in establishing the correctness of the Regge pole theory. These studies are of c r i t i c a l importance in specifying the characteristics of the quark model. - I -There i s al s o i n t e r e s t i n the use of t h i s i n t e r a c t i o n as a fundamental i n g r e d i e n t i n the study of the pion-nucleus i n t e r a c t i o n . The pion-nucleon system i s a l s o u s e f u l i n p r o v i d i n g a t e s t i n g ground f o r fundamental symmetries l i k e i s o s p i n - i n v a r i a n c e and t i m e - r e v e r s a l i n v a r i a n c e . I t i s rat h e r overwhelming that so many important comparisons between theory and experiment can be made w i t h i n the realm of pion-nucleon s c a t t e r i n g . Although a proper perspective of strong i n t e r a c t i o n physics must be acquired through the study of many hadronic systems, n e v e r t h e l e s s , very u s e f u l i n f o r m a t i o n can be obtained through the i n v e s t i g a t i o n of the pion-nucleon system alone. 1 . 1 . 2 Low energy pion-nucleon s c a t t e r i n g The pion-nucleon i n t e r a c t i o n at low energies has been the subject of numerous st u d i e s f o r many years, and i s now f a i r l y w e l l understood due to the e f f o r t s of many experimenters who have employed both pion beams and photon beams to study the various aspects of t h i s s u b j e c t . There are, n e v e r t h e l e s s , some i n t e r e s t i n g points l e f t u n s e t t l e d , and t h i s p r o j e c t addresses some of these matters. The low energy parameters from pion-nucleon s c a t t e r i n g , e s p e c i a l l y the s-wave s c a t t e r i n g lengths, are important observables f o r many strong i n t e r a c t i o n t h e o r i e s , but the experimental s i t u a t i o n has apparent i n c o n s i s t e n c i e s . F i r s t , there i s the t o p i c of low energy pion-nucleon e l a s t i c s c a t t e r i n g . One f i n d s that the experimentally a c c e s s i b l e s c a t t e r i n g process at low energies i s comprised of the e l a s t i c r e a c t i o n s : ( 1 ) u + + p + u + + p ( 2 ) TC + p •*• n~ + p - 3 -and the charge exchange s c a t t e r i n g (3) u + p - > - T t 0 + n - > - Y + Y + n ' S t r i c t l y speaking, the process (3) Is a r e a c t i o n ; however, si n c e the f i n a l s t a t e contains only two p a r t i c l e s , the same formalism as i s used to de s c r i b e e l a s t i c s c a t t e r i n g can be used. This i s why i t i s l o o s e l y r e f e r r e d to as an e l a s t i c process. Because of the d i f f i c u l t y of performing such experiments, there has been an apparent discrepancy between the s-wave s c a t t e r i n g lengths measured i n charge exchange s c a t t e r i n g d i r e c t l y , and that deduced from it p and i t + p e l a s t i c s c a t t e r i n g . This has been known f o r many years and was quoted as evidence f o r the v i o l a t i o n of hadronic i s o s p i n symmetry over and above the ± „ mass d i f f e r e n c e of the n and T C u mesons. We w i l l show that the discrepancy has disappeared, at l e a s t at low energies (ABA83), now that c e r t a i n systematic experimental e r r o r s have been e l i m i n a t e d using a b e t t e r technique. This w i l l a l s o c l a r i f y why a recent p h a s e - s h i f t a n a l y s i s by the VPI group r e s u l t e d i n an unusually high s c a t t e r i n g length (ZID80). The study of the pion-nucleon system i s not only i n t e r e s t i n g f o r i t s own sake, but al s o as the basic i n g r e d i e n t f o r studies of the i n t e r a c t i o n of pions w i t h more complex n u c l e i . O r i g i n a l l y , the low energy l i m i t of pion-nucleon s c a t t e r i n g was the t e s t i n g ground f o r PCAC and current algebra (PAG75) and i s now the subject of c h i r a l p e r t u r b a t i o n theory of QCD (GAS82). I n c h i r a l p e r t u r b a t i o n theory an expansion i s made i n powers of the quark mass around the c h i r a l l i m i t defined by zero quark masses. The quark mass term i n the QCD v 2 Lagrangian, i . e . , m, q q, = m.dd + m uu, e x p l i c i t l y breaks the c h i r a l - A -flavour symmetry SU T(2) x SU_(2). The currents of the SU_(2) x SU D(2) Lt K LI K symmetry group are represented by the weak "conserved vector" current (CVC) and "partially conserved" axial vector current (PCAC). Hence, the CVC-hypothesis and PCAC theorems appear as a natural consequence of symmetry properties of QCD. Consequently a deeper understanding of the famous relations between the axial vector coupling constant g^ of 6-decay and of the pion decay-constant f on one side, and the pion-nucleon cross sections, the TtN s-wave scattering lengths or the nN coupling constant g ^ ^ on the other side is achieved (CHE84). The determination of the o-term from the low energy pion-nucleon interaction, i . e . m — — mu + m d an N ( t = 0, v = 0) = - ^ j - < N|uu+dd|N>, where m «• ^ » plays an important N role in strong interaction physics (GAS82). As can be seen from this interaction, the very existence of a nonvanishing o-term is a consequence of the quark mass term. Hence, the a term tests the structure of the QCD Lagrangian, and i t s value determines the size of SU (2) x SU (2) symmetry breaking. The QCD prediction for the a-term i s (35 ± 5) MeV (GAS82), while the analysis of a l l available data via forward dispersion relations results in (60 ± 8) MeV (KOC82). This discrepancy between the QCD calculations and the experimental value indicates the need to perform experiments which are particularly sensitive to the value of o-term (KAR83). A l l in a l l , the existing nN data below 100 MeV are contradictory, and the results of a recent experiment from LAMPF (FRA83) has further enhanced the confusion. This experiment on u +p and it -p elastic scattering has inconsistent - 5 -normalizations. Therefore, despite the well established status of pion-nucleon phase shifts over the A(1232) resonance and beyond, there is less confidence concerning the situation below resonance, especially below 100 MeV. To provide better data for future phase shift analyses new measurements of elastic scattering and charge exchange of pions on protons are required. New phase shift analyses w i l l then f a c i l i t a t e the determination of the s-wave scattering lengths a^(isospin = 1/2) and a 3 (isospin = 3/2), as well as the even and odd isospin scattering lengths ( a + and a , respectively). New data w i l l also provide further information on the o-term mentioned earlier and the nN coupling constant. When n-mesons were f i r s t produced a r t i f i c i a l l y in laboratories, i t was soon observed that they can be obtained not only from the collisions of nucleons with nucleons, but also by bombarding nucleons with y - rays of sufficiently high energy (STE50). Four single pion production reactions are possible: (4) y + p it"*" + n (5) y + P "*• i"° + P (6) Y + n •* 1 1 + P (7) y + n ->• it 0 + n The characteristics of these reactions are dominated by the f i n a l state, thus photomeson production also studies the pion-nucleon interaction, and many experiments have been performed on a l l four reactions. The f i r s t two reactions are rather well documented because i t is relatively straightforward to define each reaction, even with the use of bremsstrahlung beams. Reactions - 6 -(6) and (7) have the obvious disadvantage that i t is necessary to use deuterium as a neutron target. This is f a i r l y satisfactory at high energies but is questionable at low energies because of d i f f i c u l t i e s in theoretical interpretation. Unfortunately, three isospin amplitudes are needed to define these reactions, so i t is essential to have data on at least three of the reactions. At low energies, i t is possible to study negative pion photo-production by investigating the time-reverse reaction (8) %~ + p •*• y + n. The advantage of pion radiative capture is that the reaction can be studied for free particles. The great disadvantage is that the radiative capture cross section is low (~ 0.7 mb) and i t is d i f f i c u l t to separate the •y-rays of this reaction from the y-rays resulting from the i t 0 decay (TT°->-2Y) i n the concurrent charge exchange reaction (o" c x varies between 5 and 50 mb) . Because the charge exchange reaction is between 10 and 60 times more p r o l i f i c in the region of A-resonance, the d i f f i c u l t y is even greater here, and i t is essential to overdetermine the kinematics. This is normally done by detecting both the y~ray and the neutron in coincidence which entails the use of two detectors, both with efficiencies much less than 100% which have to be determined independently. The radiative capture reaction has been the subject of considerable speculation in recent years. There have been suggestions that the existing data on this reaction can be made compatible with other photomeson data, only i f one postulates an Isotensor component of the electromagnetic current. Similarly, data for the reaction (n + p •*• y + n) and data for % photo-- 7 -production on deuterium exhibit considerable inconsistencies and this had led to a discussion of the possibility of a breakdown of time reversal invarlance. Another reason for a sudden reawakening of interest in the photomeson production reaction i n the resonance region is the possibility that the A has a significant fraction of d-state contribution in the quark wave function (GER81, VEN81, ISG82, BR083, DEY83, BOU86). A very accurate measurement of the angular distribution, especially from T T 0 photoproduction, i s required to determine the extent of this contribution. 1.2 Technical Advantages There have been many experiments on the low energy pion-nucleon Interaction, but the quality of data has been f a i r l y poor because of numerous technical d i f f i c u l t i e s and limitations. A major d i f f i c u l t y from the experimental point of view has been the quality of low energy pion beams. This i s principally because the flux has been low due to the d i f f i c u l t y of extracting such beams from accelerators, but i t has further been compounded by the short decay length of pions. For a pion of kinetic energy below 100 MeV, there i s a significant chance of decay en route to the experiment. This decay not only depletes the i n i t i a l beam but also adds a muon and electron contamination which can become a very serious problem. This problem was solved with the advent of meson factories. At TRIUMF, there Is the advantage of having relatively short pion beam lines, which means that i t is feasible to perform experiments down to 20 MeV while retaining a useable flux. In addition to the problem of low energy pion beams, the studies of the charge exchange reaction detected neutrons and/or y r a y s with detector of poor - 8 -resolution and uncertain efficiencies. Previous studies of the radiative capture reaction have also u t i l i z e d neutron detection which entails the estimation of counter efficiencies, a d i f f i c u l t task at the best of times, and so the errors have typically been ± 10% or more. For example, an experiment at CERN (FAV70) measured the radiative capture cross sections for pion momenta from 220 MeV/c to 380 MeV/c (T - 110 to 270 MeV). The data are for one angle only and have errors of ± 10%. Another experiment at Berkeley (BER71, COM75) by a UCLA group has investigated radiative capture reaction at momenta of 316, 450, 490 MeV/c (T - 200, 330, 370 MeV). They have measured the cross section at six angles with errors of ± 10% or worse. Both of these experiments detected the neutron and y-ray in coincidence which also requires that one determine efficiencies of detectors for neutron and y-ray to high precision. These inherent problems in experimental technique were solved in the present experiment by using the superior qualities of a large NaI(T£) detector TINA which measures 45.7cm in diameter and 50.8 cm in length. This large volume (83 l i t r e s ) i s necessary to contain the shower for y-rays in the energy range of a few hundred MeV. The ava i l a b i l i t y of this detector made i t apparent that i t would be possible to study the low energy pion reactions which produce y-rays, i.e. reactions (3) and (8). The advantage of this technique is that a single detector i s used which has a clearly defined solid angle and whose efficiency i s almost 100%, so that a more reliable cross section can be obtained with the additional advantage of detecting both reactions simultaneously. This ties together the two different reactions with consider-able confidence. Historically, this i s not the f i r s t time that a single y-ray detector has been used. In 1967 Carroll (CAR67) reported such a measurement - 9 -at 5A MeV at Berkeley, using a large plastic detector which had a resolution of = 20% FWHM. He was severely limited by statistics and obtained a very large value of Panofsky ratio for stopped n~, v i z . 1.72(8) or 1.82(8), depending on the spectral response function, whereas the present value is 1.543(8)(SPU77). Also in 1976 Bayer (BAY76) used a single Nal(Tl) photon spectrometer to measure the charge exchange differential cross section at 0° and 180°, but the resolution was not good enough. Therefore, even though this technique had been employed before, i t failed because the energy resolution of the y-ray detector was not really sufficient to distinguish the two reactions. Only recently have large Nal(Tl) crystals with high energy resolution been available. TINA attains an energy resolution of 5% for 200 MeV y~rays when a 25.4 cm collimator is used. For the highest pion energies, a 15.2 cm collimator was used which gives a slightly Improved resolution of 4%. The separation between the radiative capture and charge exchange gammas is done entirely by means of the energy resolution of the Nal detector. No attempt i s made to detect the accompanying neutrons . This greatly simplifies the experimental equipment, and removes almost a l l the uncertainties regarding detector efficiency. This experiment is thus the f i r s t to achieve a clear separation between radiative capture and charge exchange, using a single y ~ r ay detector. For the radiative capture, the alternative technique has been to study the inverse process (9) y + d •*• n~ + p + p g where p designates a spectator proton. The extraction of the elementary - 10 -cross-section is a complex procedure and i t is not possible to be confident i n this procedure to better than 10%. However, i f great care is taken the technique does provide useful information (ARG78). To summarize, because of the technical advantages that have been mentioned, the present experiment avoided several problems and enabled us to obtain data which are superior in quality and credibility and considerably improve the existing data. I.3 Charge exchange reaction and hadronlc isospin symmetry The charge exchange reaction (3) is one of the three "elastic"scattering channels for pions on protons. The others are reaction (1) and ( 2 ) . The pion nucleon interaction is strongly isospin dependent. Fig. 1.1 compares the (.TI~*~V) system cross section with (it~p) cross section at low energies. The large bump in the n +p cross section at around 195 MeV is due to the presence I | of the lowest energy pion-nucleon resonance, the A (3,3) of mass 1232 MeV. This is a wide resonance ( r «= 115 MeV), and therefore It affects the pion nucleon interaction down to much lower pion energies. It is normal to assume charge independence and thus to describe elastic scattering in terms of only two isospin amplitudes (I • 1/2) and (I «• 3/2), which means that only two experiments would be needed. However, i t is becoming more and more apparent that isospin (hadronic) is not a basic symmetry of the strong interactions, but is brought about by the similarity of the effective mass of the up and down quarks, and so i t is advisable to verify the validity of isospin invariance in as many situations as possible. To illustrate the existing - 1  -in E c to V (MeV) Figure 1.1 fit+D) and (n-p) total cross aections. The curves are calculated from pn.se . h i f t . which were fi t t e d to the world ,-nucleon data using the program SAID. - 12 -confusion, i t is worth recalling the low energy pion relations which were described by Brueckner, Serber and Watson (BRU51). They showed that one could + ± ± + relate three reactions (y + p •*• n + n), (it p + n p) and (pp •+ % d) using some experimental measurements with stopped pions, together with some basic principles such as detailed balance and isospin invariance. The relations are illustrated in Fig. 1.2 where the following abbreviations are used: C.I., Charge Independence; D.B., Detailed Balance; E.Z.E., Extrapolation to Zero Energy. The symbol o is used to designate the s-wave cross-section for i n -flig h t studies, whereas u) designates the rate of absorption for a stopped it in a mesic atom. The other quantities, eg. R, P, T, S, are experimentally determined ratios of the appropriate experimental quantities • Unfortunately the latest analysis has shown that there are discrepancies with the reaction (pp •*• it +d) at the level of 50% (SPU75), while the recent discussions of the breaking of isospin invariance have postulated effects no larger than 2%. It is clear therefore that there are some experimental problems and the Investigation of (it p) reaction at low energies to study the s-wave interaction in fl i g h t would be very helpful. Pions have unit isospin and nucleons one-half, so that the pion-nucleon system may be decomposed into total isospin states I = 1/2 and I = 3/2. The (u +p) system is a pure I = 3/2 state, while the (it p) system is a superpositon of I = 1/2 and I = 3/2 states. Hadronic isospin invariance implies that the scattering processes (1), (2) and (3) can be described in terms of a scattering matrix in isospin space, i . e . there is an I = 1/2 scattering amplitude A(l/2), associated with the scattering matrix M(l/2), and an I = 3/2 - 13 -o - ( y p — i r * n ) o - ( ^ n — 7 r ~ p ) }D.B. cr(7r"p-ny) E . 2 . E w ( 7 r " p - n y ) w ( 7 r ~ d — n n / ) I s w ( 7 r ~ d — n n ) J E . Z . E . c r ( 7 r ~ d — n n ) c r O r ' - ' d — p p ) J Q B . o - ( p p — i r + d ) c r C ^ p — 7 r * p ) JC .I . 0 " ( 7 T ~ p — n 7 T ° ) J E . Z . E . W ( 7 7 " p — n T T 0 ) Figure 1.2 The low energy pion-nucleon relations of Brueckner, Serber and Watson. - 14 -scattering amplitude A(3/2) associated with the matrix M(3/2). These matrices are completely independent, and the scattering amplitudes can have quite different values. The cross section for these processes can be written as: ff+Or+p ir +p) a |A(3/2)|2 (l.a) o~(ir~p •»- T T " p ) a 1/9 |2A(l/2) + A(3/2)| 2 (l.b) O C X ( T T " p Tr°n) a | |A(l/2) - A(3/2) | 2 (l.c) It i s known that the requirement of isospin invariance leads to so called "triangle" relations (KAL64) which connect differential cross sections of ± ± -(TT p) elastic scattering (a ) and (IT p) charge exchange scattering (°"cx) as follows: (/o+ - /a") 2 < 2 a c x < (•a+ + /a") 2 (2) These inequalities define limits for o"cx i f the values of a + and a are known and hence constitute a test of isospin invariance. 1.4 Radiative capture and general study of photomeson reactions: There are four reactions in which single pions are photoproduced by nucleons, v i z . reactions (4), (5), (6), and (7). These reactions can be described by three amplitudes which relate the i n i t i a l and f i n a l hadron states of specified isotopic spin (the electromagnetic interaction, of course, does - 15 -not conserve Isospin). There is one isoscalar amplitude and two isovector amplitudes, and we assume for the moment that there is no isotensor component. The isoscalar and isovector amplitudes are defined as follows: S,(AI = 0) for <I = 1/2 | S | I - l/2> transition, V 1(AI = 0) for <I - 1/2 | V | I - l/2> transition, and V 3(AI = 1) for <I = 3/2 | V | I = l/2> transition. where, as i t was mentioned, the isotopic spin states refer to the hadronic system, i.e. the nucleon in the i n i t i a l state and the pion and nucleon In the fi n a l state. The photomeson production reactions can be expressed In terms of these amplitudes (DON78): yp -• w°p : (1/3 Vl + 2/3 V3 + S^, YP + n +n : /2(l/3 Vx - 1/3 V3 + S1 ), yn n°n : (1/3 Vx + 2/3 V3 - S^, yn •*• w~p : /2(-l/3 Vj + 1/3 V3 + 8 ^ . Therefore to obtain complete experimental Information, one needs detailed information on at least three of these reactions. The f i r s t two processes have been intensively studied using hydrogen targets. There have been some experiments on photoproduction with neutrons, using deuterium targets, but there are uncertainties In extracting the free neutron data because of the "spectator" proton, which is rather active, and an accurate estimation of i t s - 16 -effects, especially at low energies, is quite d i f f i c u l t . Von Holtey has given a review of the experimental situation in the f i r s t resonance region (VON73). It has also been studied by Argan et a l . (ARG78) in the A(1236) region using well defined kinematic conditions. Using detailed balance one can relate (n p -*• Y n) a°d (Y n •* 1 1 P)» therefore the radiative capture reaction offers the possibility of determining the isoscalar amplitude unambiguously. Calculations of these amplitudes have been performed using dispersion relations, but there are many discrepancies with existing data even for photoproduction on protons (e.g. the asymmetries .for polarized photons). The Interest in low energy photomeson production was revived from analyses by Sanda and Shaw (SAN71) of a l l the existing data in the f i r s t resonance region, i . e . from threshold at = 145 MeV up to 500 MeV. They showed that the present data were not internally consistent but could be reconciled i f one postulated an isotensor component to the electromagnetic current which violates C-invariance (charge symmetry). This postulate implies a violation of time-reversal invariance i f one believes in the CPT theorem and in parity invariance in the electro-magnetic interaction. This analysis was carried further by Donnachie and Shaw (D0N71) who looked for direct evidence for time reversal invariance by comparing the reactions (n p •*• y n) a n d (Y n 1 1 P)* They showed that indeed there was evidence for a difference between the two. Other analyses have argued that there is no strong evidence for an isotensor component (N0E71, BER71, DEV72), although Donnachie and Shaw (DON72) continued to believe that their original analysis was correct. Several experiments claimed to be incompatible with the isotensor current, and the interpretation that Blackwenn - 17 -et a l . (BLA72) put on their data has been c r i t i c i z e d by Shaw. Another experiment (FUJ72) has performed the "dip test" and found no effect. This was even more unsettling than the Blackwenn results and there was considerable disagreement amongst photoproduction data for the dip test. Discussions of the existing photoproduction data continued to be published (AZN72), with the general consensus that something was wrong, either the data or the basic assumptions made in analysing them. The hypothesis of partial conservation of the axial current also makes specific predictions (DEL66, ERI67, PIE67) for radiative capture of pions at low energies. Recently, there has been a sudden interest in the photomeson production reactions in the resonance region because of the possibility that the A is composed of a significant fraction of d-state contribution to the quark wave function (GER81, VEN81, ISG82, BR083, DEY83, JUR83, BOU86). The problem i s that there have been suggestions that the d-state contribution might be very large and make i t s e l f f e l t by a significant E2 transition in the A resonance, whereas the experimental evidence to date i s that the E2 multipole Is quite modest (CRA80, ARA80). The best evidence is probably from ir°-photoproduction which exhibits a very symmetrical angular distribution about 90°. The disadvantage of i r + and TT photoproduction is that the resonance contribution i s contaminated by a photoelectric term which confuses the situation. Nevertheless, accurate angular distributions for reaction (8) would be useful additional information. 1.5 A Review of the Previous Work Previous published experiments of the ( I T p -»• yn) reaction were done - 18 -mostly in the time reversed channel (ADA69, BEN73, FUJ77, ARG78, FAU84) with the corresponding problem of deuterium targets. Prior to these experiments, results had differed by as much as a factor of 2. At low energy there have been several investigations of the radiative capture reaction (BER74, BAL77, C0M75, GUE75, TRA79). Even the best of these experiments has an error of 10% or higher. The f i r s t stage of analysis of these data i s a multipole analysis, and these are partly discussed by TRAN et a l . (TRA79). There is also continuing effort at Glasgow (BAR78, CRA80, CRA83) and Tokyo (ARA80, ARA82). Because of the lack of polarization data for photoproduction, these calculations normally obtain the imaginary part of the multipoles from existing pion-nucleon phase-shifts, via the Fermi-Watson theorem. A recent attempt has been made by Smith and Zagury (SMI79) to use only photoproduction data, but i t was not very successful because of the limited data base which i s available. For the charge exchange channel, there were several early investigations at low energy (SPR54, MIY62, R0B54, CUN65, D0N66, FRA67), but as i t has already been indicated, the total cross sections were higher than the predictions from phase-shift analyses which were based mainly on elastic scattering data. A more recent experiment at Saclay (DUC73) measured the charge exchange cross-section at 180° for pion kinetic energy, T^, of 22.6, 32.9, and 42.6 MeV. Using p-wave phase shifts from analyses existing at that time, they obtained an s-wave scattering length more compatible with the elastic scattering data, although there i s an energy variation in their scattering lengths. Above 100 MeV there i s less controversy because of the accurate results of Bugg et a l . (BUG71) as well as the less accurate, but - 19 -consistent, data from the Lausanne-Munich-CERN collaboration (JEN77) and the UCLA experiments at LBL (BER72, COM75). There are several pion-nucleon phase-shift analyses and the low energy results are discussed in the review by Nagels et a l . (NAG76). We w i l l compare our results with the most recent versions, v i z . those of the VPI group (ZID80, ARN85), and the analysis by Koch and Pietarinen, normally termed the Karlsruhe set (KOC80). These analyses are f a i r l y consistent over the A(1232) resonance, because of the quality of the data in that region, but they diverge at low energies because different selections of the data have been made. The published VPI analysis of Z i d e l l et a l . (ZID80) predicts a higher charge exchange cross-sections at low energy (~ 7mb), while a more recent VPI analysis E-85 (ARN85), as well as the Karlsruhe set, predict a value of about 5mb, which is consistent with most recent analyses. This consistency is satisfying because recent discussions of isospin breaking in the pion-nucleon interaction suggest that the effect w i l l be small, i f i t is simply due to the known Coulomb interaction plus the mass difference of the "up" and "down" quarks. (TOR72, ISG79, ISG80, DAV81). CHAPTER II  The Experiment II .1 Characteristics of low energy pion beams Because the average pion l i f e time i s only 26ns in i t s rest frame, pion beams must be produced as secondary beams. Usually an energetic beam of protons is focussed on a pion production target and the nuclear interactions in the target produce a wide spectrum of pions of different energies. A system of magnets could be set up close to the production target to collect the pions of any desired energy. Since bending magnets select particles of a given momentum and near the production target there are many different particles with a l l possible momenta, pion beams are usually contaminated by other particles. These contaminants are usually protons, neutrons, muons and electrons. Pions which decay in flig h t produce another source of beam contamination. For example, 60 MeV pions travel at a speed of 0.71c and In the laboratory frame their l i f e time is 37ns. This means that approximately 12% of the pions in the beam decay to muons for each meter of flight path. The Jacobian of the transformation from the CM. to the laboratory frame exhibits a cusp which is shown in Fig. II .1. The maximum angle at which decay muons appear is given by * a 0. 271 tan 9 = ™ a x Y ( P 2 - 0.0737) - 21 -T i i 2 0 0 Jocobion Transformation for 60 MeV decoy ^ 1 0 0 a J I I L J i_ 10 L a b A n g l e ( O e g ) I S Figure II.1 The Jacoblan of transformation from the rest frame of a 60 MeV pion to the Lab frame. Since the angular distribution of the decay muon Is Isotropic In the re6t frame of the pion, the angular distribution of the decay muon In the Lab would also have the same shape as the figure. - 22 -where 6 is the pion speed in multiples of the speed of light and y = (1 - 8 2 ) ~ 1 / 2 . For 60 MeV pions 8 =16.2 degrees. Therefore one max ° expects to find a large number of muons at an angle close to 6 . This i s max why low energy pion beams are surrounded by a halo of muons, which make i t very d i f f i c u l t to perform experiments at extreme forward angles. II .2 The Pion beam line at TRIUMF The experiment was performed at the TRIUMF f a c i l i t y . The accelerator has no macroscopic duty cycle, but the beam has a micro-structure consisting of a beam burst every 43 ns caused by the RF of 23.055 MHz. Negative hydrogen ions are injected into the center of the cyclotron and accelerated up to 500 MeV, where they are stripped of their electrons, and the resulting proton beam, of up to 130 uA, is extracted out of the accelerator tank and directed upon the pion production target T l . The usual production targets are made of carbon or beryllium. The pions thus produced are guided down a secondary channel called M i l . The TRIUMF Mil channel is designed for production of rt + and u over an energy range from 50 to 350 MeV. Its design i s such as to provide a high flux, high resolution channel with moderate momentum and solid angle acceptances. The layout of the Mil pion channel is shown in Fig. II .2. A description of the optics of the channel has been given in the design note by Stinson (STI80). An interim report on the performance of Mil was given by Aniol (ANI81). For the present experiment, the secondary beam was set for negatively charged particles with a series of magnets. Therefore TI , p, , and e were a l l present in the pion beam, but there were no protons, which are a - 23 -Figure II.2 The layout of Mil pion channel. - 24 -serious problem in positive beams. The channel has a momentum acceptance of Ap/p = 2%. The negative pions from the channel were used to bombard a liquid hydrogen target. II .3 The Liquid Hydrogen Target The liquid hydrogen target was contained in a flask 14 cm in diameter and 4.40 cm thick with f l a t mylar walls, surrounded by hydrogen gas in pressure equilibrium with the l i q u i d . A guard ring was used to stop bubbles from traversing the target volume. The target assembly was contained in a vacuum chamber with a .079 cm thick mylar window. The total thickness in the beam path was 0.110 g/cm2 of mylar. The target thickness was measured to be 4.47 cm and 4.40 cm for the forward and backward targets, respectively. The angles of these targets with respect to the beam were 31° and -29.5°, as measured with a laser beam across fid u c i a l marks on the targets. During the experiment, the targets were kept cool by a refrigerator controlled by the pressure at the flask, allowing a maximum variation of temperature between 20.5K and 20.7K. Therefore, the density of the liquid hydrogen (0.0703 g/cm3) changed less than 1%. The total uncertainty of the target proton number was 1.5%. The hydrogen target was cooled by a CTI (Cryogenic Technology, Inc., of Waltham, Mass.) cryodyne helium refrigerator (model 1020). This unit has a two stage cooling cycle with the f i r s t stage cooling the thermal radiation shield of the condensor, while the second stage cools the H 2 gas to liquid temperatures. The f i r s t stage could handle a 25 watt heat load at 27K, while the condensor was able to handle a 10 watt load at 20K. By thermally - 25 -isolating the target flask in a vacuum and surrounding the flask with layers of aluminized mylar sheets, the heat load on the condensor at equilibrium was kept to about 1 - 2 watts. I n i t i a l l y there were some problems with liquid f i l l i n g the gas buffer region, so s l i t s were cut in the aluminized mylar so that the level in the flask could be observed. This also increased the heat load on the gas region, which was useful. The target was run slightly above one atmosphere, in order to avoid the possibility of leakage of air into the hydrogen target. II .4 The Nal(TA) Detector II .4.1 Characteristics of Alkali Halide Scintillators The discovery of thallium-activated sodium iodide In the early 1950's began the age of modern s c i n t i l l a t i o n spectroscopy of y - r a y s • It Is remarkable that the same material remains pre-eminent in the f i e l d , despite nearly three decades of research into other s c i n t i l l a t i o n materials. Large ingots can be grown from high-purity Nal, to which about 10 - 3 mole fraction of thallium has been added as an activator. These impurities, or "activators", create special sites in the lattice (Fig. II .3) at which the normal energy band structure is modified from that of the pure crystal. As a result, there w i l l be energy states created within the forbidden gap through which the electron can de-excite back to the valence band. Because the energy is less than that of the f u l l forbidden gap, this transition can give rise to a visible photon, and therefore serve as the basis of the s c i n t i l l a t i o n process . Another important consequence of luminescence through the activator sites Is - 26 -Conduction band Activator excited states Band J Scintillation j gap 1 photon v Activator ground state Valence band Figure II.3 Energy band structure of an activated crystalline s c i n t i l l a t o r . - l i -the, fact that the crystal can be transparent to the s c i n t i l l a t i o n l i g h t . In the pure crystal, roughly the same energy would be required to excite an electron- hole pair as that liberated when that pair recombines. As a result, the emission and absorption spectrum w i l l overlap and there w i l l be substantial self-absorption. Howeverj the emission from an activated crystal occurs at an activator site where the energy transition i s less than that represented by the creation of the electron-hole pair. Therefore, the emission spectrum is shifted to longer wavelengths and w i l l not be influenced by the optical absorption band of the bulk crystal. Scintillators of unusual size or shape can also be fabricated by pressing small crystallites together. Nal(TJl) is hygroscopic and w i l l deteriorate due to water absorption i f exposed to the atmosphere for any length of time. Crystals must therefore be "canned" in an air-tight container for normal use . The most notable property of Nal(TJl) is i t s excellent light yield, which is the highest of any known s c i n t i l l a t i o n material to primary or secondary electrons. Its response to electrons and y-rajs is close to linear. It has come to be accepted as the standard s c i n t i l l a t i o n material for routine y-ray spectroscopy, and can be machined into a wide assortment of sizes and shapes. However, the crystal i s somewhat fragile, and can easily be damaged by mechanical or thermal shock. The dominant decay time of the s c i n t i l l a t i o n pulse is 230 ns, uncomfortably long for some fast timing or high counting rate application. In addition to this prompt yield, a phosphorescence with characteristic 0.15 sec decay time has been measured (KOI73), which - 28 -contributes about 9 percent to the overall light y i e l d . Other long-lived phosphorescence components have also been measured (DEL68). II .A .2 The TINA Detector For this experiment, the most important piece of equipment was a large Nal(TI) crystal called "TINA" (for TRIUMF Iodide of Natrium). The crystal measured 45.7 cm in diameter by 50.8 cm in length and was grown in two units. It was manufactured by the Harshaw Chemical Company, Ohio, U.S.A., and is used for the detection of high energy y r a y s and electrons around a few hundred MeV. The crystal i s hermetically sealed in an aluminum can. TINA is a crystal of excellent quality and can achieve an energy resolution of 4% at 130 MeV with a small collimator, which is as good as has been obtained with such a device. Such detectors are preferred when 100% efficiency, high resolution and a reasonably short decay time are important characteristics. The TINA crystal is an optically single unit viewed by seven RCA 4522 phototubes . The phototubes have a diameter of 12 .7 cm and a risetime of about 3ns at maximum voltage of 3000V. However, the rise time of the summed pulse is about 50ns, due to variation in pathlength that the light must travel before i t reaches the phototubes. While the use of several phototubes to view the light avoids the problem of non-uniformity of the photo sensitive area encountered in a single large phototube, and hence gives a better energy resolution (HAS74), the collection of tubes must be balanced to effectively make use of this advantage. Balancing the phototubes was made relatively simple for this experiment by the prominance of the y~rays from stopped pions. By connecting - 29 -the phototubes i n d i v i d u a l l y at the f a n - i n and va r y i n g the high voltage of each, the phototubes could q u i c k l y be balanced or rechecked. The a c t u a l detector response at a given energy depends on a v a r i e t y of f a c t o r s , i n c l u d i n g : 1) The area of the c r y s t a l ' s f r o n t face which p a r t i c l e s are permitted to ente r , i . e . the f r o n t face i l l u m i n a t i o n ; 2) the count r a t e i n the c r y s t a l ; and 3) the accuracy w i t h which the tubes have been balanced. I n b r i e f , the N a l ( T l ) c r y s t a l was e s s e n t i a l l y 100% e f f i c i e n t and had s u f f i c i e n t r e s o l v i n g power to separate the y-rays from r a d i a t i v e capture and charge exchange r e a c t i o n s . Furthermore, the timing r e s o l u t i o n of <3.0 nsec (FWHM) obtained w i t h a constant f r a c t i o n d i s c r i m i n a t o r made i t p o s s i b l e to separate the y~rays from the la r g e neutron background which v a r i e s i n energy up to a maximum of about 80 MeV (the f l i g h t path of about lm gives 6ns time separation between y-rays and the high energy neutrons). The energy r e s o l u t i o n of Tina (4% at 130 MeV under normal c o n d i t i o n s ) was obtained from the width of the r a d i a t i v e capture y~rays of stopped pions i n l i q u i d hydrogen, a convenient monochromatic source at 129.4 MeV. I I .5 The Experiment 11.5.1 Set-up and event d e f i n i t i o n Negative pions from the M i l channel at TRIUMF were used to bombard a l i q u i d hydrogen t a r g e t . The incoming beam was defined and counted by a t h r e e -- 30 -plastic-scintillation-detector telescope S1.S2.S3. The experimental setup i s shown in Fig. II .4. An intensity of approximately 105 pions per second was used with beam energies of T = 50 MeV, 66 MeV, 80 MeV, 95 MeV, 110 MeV and 125 MeV. The y~ rays from the Tt + p •*• y + n and i t " + p ->• Tt° + n (T I 0 -»• 2y) reactions were detected in a 46 cm diameter and 51 cm long Nal(Tl) crystal (TINA) at laboratory angles from 30° to 140°. The main task was to separate the y~ rays from these two different reactions on the basis of energy resolution. Kinematics of the reactions Is shown in Table II.1. The y-rays from the n° decay can have quite a low energy (down to 15 MeV) so that in this low energy region the neutrons can be troublesome. To solve this problem, we used time of flight technique to separate neutrons and Y~ray signals in the Nal(Tl) crystal. In order to avoid interference from the walls of the target vessel, separate liquid hydrogen target cells were used for measurements in the forward and backward directions. A s c i n t i l l a t i o n counter S4 in front of the v-detector was used to identify the charged particles which entered the crystal. Events defined as a coincidence of (SI. S2. S3. S4 TINA) were recorded on tape during the course of experiment with the aid of a PDP11/34 computer. For each event the amplitudes of the signals in S3 and TINA were recorded, together with timing signals from the primary proton beam, the S3 counter, and TINA. These timing signals were used to identify pions i n the beam and y~rays in TINA via time of flight technique. The logic signal of counter S4 was also recorded and used to reject the charged particles in TINA later during the data analysis. The beam size and position at the target was determined using a multi-wire proportional counter. The beam spot was a - 31 -Figure II.4 Experimental set-up. SI, S2, S3 and S4 are s c i n t i l l a t i o n counters. TINA Is a NaI(T£) y-ray detector, movable around the target from 0° to 145°, and Is surrounded by an Iron shield 25 cm thick at the front and 10 cm thick on the sides. TABLE II.1 Kinematic of the Reactions (n~p •*• yn) and ( T I ~ P + Tt°n •*• y + y + n) Pion Energy (MeV) 0 l a b E Y (MeV) (MeV) n i n a x Kny (Mev) A ( % ) 50 40° 181 49 155 14 50 130° 154 31 131 15 100 40° 227 94 207 9 100 130° 179 56 163 9 200 40° 314 181 302 4 200 130° 222 99 211 5 Notes: 1) E^ is the gamma-ray energy from (n~p yn). 2) E*?fx is the maximum gamma-ray energy from the decay of the n° from 1 lY (n~p -»• Tt°n) . 3) A is the minimum difference between true events and the ii0 background - 33 -horizontal ellipse, 1.5 cm wide and 1.3 cm high, which increased to 2.84 cm by 2.43 cm when the beam telescope was in place. The beam defining counter S3 was placed 20.5 cm upstream of the target center. The beam composition was determined from the time of flight spectra of the relative time of the primary proton beam and the secondary pion beam at S3. A typical time spectrum of the beam is shown in Fig. II .5. The fractions of Tt:u;e at different beam energies are also shown in Table II .2. The acceptance of y-rays into the Nal(Tl) detector was defined by a lead collimator, with an aperture 25.4 cm In diameter for beam energies below 90 MeV, and an aperture 15.24 cm in diameter for beam energies greater than 90 MeV. The collimator was placed in front of the crystal, with the back edge of the collimator 95.6 cm from the target center. The mean beam energy at the target center was determined by correcting the incident pion beam energy for energy losses in the beam counters and target windows as well as half of the target l i q u i d . The incident beam momentum was determined using a previous calibration with an a-source and measurements of the bending dipoles magnetic fields with NMR probes. From these measurements, the mean pion beam energy at the center of the target was determined and the result is shown in Table II .3. The momentum acceptance of the channel was — = 2% for a l l energies, equivalent to P a corresponding beam energy spread which is also shown i n table II .3. At each angle of the y-ray detector several runs were recorded with the target f u l l and empty. The empty target runs were used for the subtraction of background from the target windows and vacuum container walls. During the data taking, several runs were performed with stopped pions by introducing a degrader - 34 -7 14 21 28 35 TIME (ARBITRARY UNITS) Figure II.5 The time distribution of the beam particles (S1.S2.S3) with respect to the IF buckets of the primary proton beam (T^ - 66 MeV). The beam composition at S3 is obtained from this histogram. - 35 -TABLE II.2 Pion Beam Fraction at Different Energies Pion Beam Energy (MeV) Pion Fraction 50 0.31 ± 1% 66 0.49 ± 1% 80 0.63 ± 1% 95 0.76 ± 1% 110 0.83 ± 1% 125 0.88 ± 1% - 36 -TABLE II.3 Pion Beam Energy and Energy Spread Beam Energy (MeV) Beam Energy in the center of target (MeV) Beam Energy Spread (MeV) 50 45.6 ± 2 % 0.9 66 62.2 ± 2 % 1.2 80 76.4 ± 2 % 1.5 95 91.7 ± 2 % 1.8 110 106.8 ± 2 % 2.1 125 121.9 ± 2 % 2.4 - 37 -between S2 and S3. This provided a good spectrum of y-rays from stopped pions for energy calibration and the test of the response function of TINA. II.5.2 Plastic S c i n t i l l a t i o n Counters The charged particle detectors were made from plastic s c i n t i l l a t o r NE110, manufactured by Nuclear Enterprises. These identified the incoming pions, the scattered protons and pions, and also were used for rejecting the charged particles which entered Tina. The counters were viewed by RCA8575 phototubes. Since the plastic counters supplied most of the timing informations, care was taken to make sure that a l l counters were functioning properly. The S3 counter placed the tightest constraint on the incident pion's trajectory, and it s small size was chosen in order to make sure that the pion hit the hydrogen flask. Dimensions and efficiencies of the counters used in this experiment are given in Table II.4. II .5 .3 Electronics Figure II .6 shows the schematic of the electronics. The electronics may be divided into 5 categories: 1) Identify a negative pion in the liquid hydrogen target; 2) Determine the energy of the photon; 3) Determine the time of fli g h t of the photon; 4) Process data in PDP11/34 computer to produce on-line information; 5) Record data on tape for off-line analysis. An event was defined as SI. S2. S3. S4. Tina coincidence. If a - 38 -TABLE II.4 Dimensions and Efficiencies of Plastic S c i n t i l l a t i o n Counters Counter Size (cm) Efficiency SI 10 x 10 x 0.16 99.8 S2 7.6 (dia) x 0.16 99.9 S3 (defining counter) 6.4 (dia) x 0.16 99.9 S4 (veto counter) 48.3 x 48.3 x 0.32 99.9 - 39 -Figure I I . 6 Schematic diagram of electronic!. - 40 -coincidence i s found, the anode and dynode signals from Nal(Tl) are amplified, digitized and histogrammed. The dynode signals were summed and amplified i n the experimental area, which makes i t less vulnerable to the noise pickup via the cable. The dynode signals were subsequently digitized using a Northern ADC. For processing the anode signals of the phototubes viewing the crystal, signals from each of the seven tubes were brought independently from the experimental area to the counting room via low loss cables. The signals were then combined using an LRS active fan-in. The summed pulse was clipped to approximately 300 ns, which reduced pile-up effects while introducing essentially no deterioration in the energy resolution. The summed signal was then fanned out to allow checks to be made without interrupting data acquisition, and then passed through a linear gate and into an amplifier, which integrated the signal for a time constant of 1 p,s. The pulse height was digitized using an LRS2259 Camac ADC. So, the occurence of an event resulted in the clipped anode pulse and the dynode pulse of the summed Nal(Tl) signal being recorded in an LRS2259 Camac module ADC, and a Northern ADC respectively. Timing signals were also derived from the anode signals. An Ortec 934 constant fraction discriminator (CFD) was used on the Nal pulse to time the arrival of neutral particles . The timing resolution of 2.0 nsec (FWHM) obtained from CFD made i t possible to separate the y ~ r a y s from the large neutron background via the time of flight technique. The flight path of about one meter gives 6 ns time separation between Y~rays and the high energy neutrons. This time distribution was obtained with a "stop" signal from the - 41 -SI. S2. S3 coincidence and a "start" signal from the crystal. The time of fli g h t spectrum of neutral particles i s shown in Fig. II.7. Both camac and visual scalers were used during the experiment. There was a camac scaler which was set i f the S4 counter fired, thus indicating whether the particle which had entered the crystal was charged or uncharged. This scaler was reset to zero after each event. The event rate and the SI. S2. S3 coincidence rate were recorded in camac scalers, and these scalers were operative as long as the computer was free to accept events, otherwise they were inhibited. The computer read the camac modules, and processed the data to form the histograms. During this read-time interval the camac crate was inhibited, but the data taking rate was sufficiently slow, so that this did not seriously affect the experiment. In fact the on-line histograms were quite useful and necessary for understanding the progress of the experiment. II.5.4 The Test Run Before starting this project, a test run had been performed using the low energy pion channel M13 at TRIUMF. The purposes of the test run were severalfold, among them: 1) To determine the percent resolution of each ADC as a function of electron energy. Four different ADCs have been tested (Northern 8192, Lecroy 2259A, 2249W, 2249A); 2) To determine which ADC is the best for electrons; 3) To determine whether to use a dynode balance or anode balance for the Nal photomultiplier tubes; - 42 -O CO »-z 3 O u 10. 7.5 * 62 MeV TIME OF FLIGHT S3 - TINA 2.5 n 20 25 30 35 40 TIME (ARBITRARY UNITS) 45 50 Figure II.7 The time distribution of neutral events in the Nal(TA) crystal, obtained by a "start" signal from S1.S2.S3 coincidence and stop signal from the crystal (T^ - 62 MeV). - 43 -4) To determine the percent resolution of each ADC for the monoenergetic 129.4 MeV y-rays; 5) To find which ADC is the best for y-rays. Based on the results from this test run which are shown in Figs. II .8 and II.9, we have chosen Northern 8192 and LRS2259 ADCs. Also the experiment indicated that a dynode balance produces the best energy resolution. - 44 -Pe+(MeV/c) 1 2 0 Figure II.8 Results of the test run for different ADCs using an e + beam. Ey * 129.5 MeV Dynode Balance O 6M Collimator • 10" Collimator A 12" Collimator Figure II .9 Results of the test run for different ADCs using stopped pion gamma ray. - 46 -CHAPTER III  Data Acquisition and Analysis I I I . l Data Acquisition The TRIUMF data acquisition program, DA, and the analysis program, Multi (FER79), running under the operating system RSX11 on a PDP11/34 computer, was used to read camac modules, write data onto magnetic tape and perform online analysis. After an Interrupt from camac, the computer suspends any analysis of the previous event which may have been in progress, reads a l l the defined camac modules according to a specification f i l e called camac. def., f i l l s the "small buffer", and passes the data to the "big buffer". When there is not enough room lef t in the buffer for an event, the scalers are read and the buffer written onto magnetic tape as one block. Then, the data is passed buffer by buffer to Multi, which has a lower priority than a DA task. A useful feature of Multi is that the online analysis program may be modified and enhanced while s t i l l taking data. For our experiment an event was defined by the coincidence SI. S2. S3. S4. Tina. This coincidence strobed the camac coincidence buffer pattern unit which then generated a LAM (a camac interrupt called "look at me"), interrupting the computer. A l l of the scalers and the event trigger were gated off from the time of an event strobe un t i l the computer had finished reading a l l of the camac modules and was ready to accept another event. The gates or vetos to the various modules came from a logic fan in/out, which acted as a logic or gate, giving an output NIM level, as long as any input has - 47 -a NIM pulse applied. The f i r s t input to arrive at this module was the event strobe. The output then started a gate generator which was adjusted to provide a pulse long enough to overlap the beginning of the "computer busy" signal from an input register. The event strobe pulse was long enough to overlap the start of the pulse from the gate generator. In this case a continuous "busy" gate was provided u n t i l the computer was ready to accept another event and removed the signal from the output register. I l l .2 Data Analysis III.2.1 Introduction The data were analysed on the VAX 11/780 at TRIUMF. Histogramming of the data was performed with the program Molli (BEN82) and the subsequent histograms stored on magnetic tape. These histograms were then displayed and analyzed with the program Replay (BEN82), and the f i n a l results were written into f i l e s and saved on disk as well as on magnetic tape. A computer program called Minuit (MIN68), which was developed at CERN, was used for the chi-squared minimization. The term "cuts" in this discussion is used to signify the selection of data which have measured parameters within certain limits (position of cuts). Because the data was recorded on magnetic tape, i t was possible to generate various histograms and scatterplots which were not available in the online analysis and to adjust the definition of y-rays, on the basis of the neutral time-of-flight spectra, in the case of a time-walk. - 48 -III.2.2 The Response Function Of the various ways y-rays can interact in matter, only three interaction mechanisms have any real significance in y-ray spectroscopy: photoelectric absorption, Compton scattering, and pair production. Photoelectric absorption predominates for low energy y-rays up to several hundred keV, pair production dominates for high energy y-rays (above 5-10 MeV), and Compton scattering is the most probable process over the range of energies between these extremes. For the case of a sufficiently large detector, like TINA, a l l secondary radiations, including annihilation photons, also interact within the detector active volume and few escape from the surface. It is interesting to see how the large size of the detector simplifies i t s response function. Some typical histories of a shower, obtained by following a particular source y-ray and a l l subsequent secondary radiation, are shown in Fig. I I I . l . If the i n i t i a l interaction is a pair production event, the electron-positron pair w i l l subsequently interact at some other location within the detector. This second interaction can be a Compton scattering or may be a pair annihilation, which produces annihilation photons when the positron is i n f l i g h t or stopped, I.e. secondary photons of s t i l l lower energy are produced. Eventually, a photoelectric absorption may occur and the history is terminated at that point. The primary and secondary y-rays travel at the speed of light In the detector medium. If the average migration distance of the secondary y-rays is of the order of 10 cm, the total elapsed time from start to f i n i s h of the history w i l l be less than a nano second. Therefore the net effect is to create electrons at each scattering point and the f i n a l photoelectron in - 4 9 -Figure I I I . l The "large detector" extreme in gamma ray spectroscopy. A l l gamma ray photons, no matter how complex their node of Interaction, ultimately deposit a l l their energy In the detector. Some representative histories are shown at the top (KN079). - 50 -time coincidence. The pulse produced by the detector w i l l therefore be the sum of the response due to each individual electron. If the detector responds linearly to electron energy, (which is not actually the case), then a pulse i s produced which is proportional to the total energy of a l l the electrons produced along the history. Because we assumed that nothing escapes from the detector, this total electron energy must be approximately the original energy of the Y~ r ay photon, no matter how complex any specific history may be. The detector response i s , therefore, the same as i f the original y-ray photon had undergone a simple photoelectric absorption in a single step. The conclusion to be reached is therefore: If the detector is sufficiently large (many tens of cm) and i t s response linearly dependent on electron kinetic energy, then the signal pulse is identical for a l l y-ray photons of the same energy, regardless of the details of each individual history. This circumstance is fortunate because the detector response function now consists of a single peak. The a b i l i t y to interpret complex y-ray spectra Involving many different energies, as i s the case in this project, is obviously enhanced when the response function consists of a single peak which Is called the f u l l energy peak. One of the mechanisms by which secondary electrons lose energy i s by radiation of bremsstrahlung photons. The fraction of energy lost by this process increases sharply with electron energy, and becomes the dominant process for electrons with energy over a few MeV. Even though the electron i t s e l f may be ful l y stopped within the detector, there is a possibility that some fraction of the bremsstrahlung photons may escape without being reabsorbed. This leakage w i l l tend to distort the response - 51 -function by moving some events to a lower amplitude from that which would be observed i f the entire electron energy were collected. This escape i s most important when the incident y-ray energy is large. For both secondary electrons or bremsstrahlung escape, the effects on the response function shape are relatively subtle. In contrast to the theoretical energy deposition spectrum, the measured response function contains the "blurring" effects due to various effects, including the f i n i t e energy resolution of the detector. The most striking difference i s the fact that a l l peaks now have some f i n i t e width rather than appearing as narrow, sharp lines. The f i n i t e energy resolution of any detector may contain contributions due to the separate effects of photon and photoelectron s t a t i s t i c s , electronic noise, variations in the detector response over i t s active volume, and drifts in operating parameters over the course of the measurement. Among them, s t a t i s t i c a l spreads are the single most important cause of peak broadening in small Nal s c i n t i l l a t o r s . Also, the response function to be expected for a y-ray detector w i l l depend on the size, shape, composition of the detector, and the geometric details of irradiation conditions. In general, the response function is too complicated to predict in detail other than through the use of Monte Carlo calculations which simulate the histories actually taking place in a detector of the same size and composition. The response function of Tina was determined by using the Monte Carlo calculations and by measuring the response of the detector to monoenergetic y-rays (129.4 MeV) resulting from negative stopped pions captured in liquid hydrogen. The f i n a l choice was an error function distribution which agreed quite well with the measured - 52 -asymmetric response function of the detector. The nature of the asymmetry was such that the half width related to the high energy t a i l of the f u l l energy peak was smaller than the half width related to the low energy t a i l . The measured response functions of TINA for two different collimators were f i t t e d with error function distribution and other candidates. For each collimator, the error function response produced the best f i t . Therefore, the following response function was chosen: where A is the amplitude, B is the peak position, and C and D are half width related to high and low energy t a i l of the f u l l energy peak respectively. A plot of the response function i s given in Fig. I l l .2. Also the f i t to the stopped pion spectra (solid line) i s shown in Figs. I l l .3 and III .4. The parameters C and D, which determine the resolution, are energy dependent. Their energy dependence was determined from Monte Carlo calculations and from the f i t t i n g to stopped pion spectra. Table I I I . l illustrates the result of Monte Carlo calculations and demonstrates how the parameters C and D vary with energy. After careful comparison of the experimental results and theoretical calculations, i t was decided that there were two possible ways to describe this energy dependence (the available information was insufficient to decide between them): P(EY,A,B,C,D) - A exp ((E Y - B/D)(l - erf ( 5 * _ 0) 1) C = C 0/E Y D = DQE Y Figure III.2 A plot of the response function of the NAI(TJl) detector. E (MeV) Figure III.3 The stopped pion gamma ray spectrum in TINA using a 25.4 cm collimator. T i 1 1 1——— 1 R STOPPED PION SPECTRUM E (MeV) Figure III.4 The stopped pion gamma ray spectrum in TINA using a 15.24 cm collimater. - 56 -TABLE I I I . l Results of Monte Carlo Simulation for Response Function Parameters C and D Y = A Exp (^B-) [ l - ERF (^-)] Ey (MeV) C(MeV) D(MeV) D/C R(%) 50 1.52 ± 0.05 1.43 ± 0.07 0.94 ± 0.06 5.90 100 2.72 ± 0.08 2.09 ± 0.10 0.77 ± 0.04 5.0 130 3.35 ± 0.1 2.48 ± 0.12 0.74 ± 0.04 4.48 200 4.27 ± 0.08 3.26 ± 0.17 0.64 ± 0.04 3.77 300 6.49 ± 0.21 4.39 + 0.25 0.68 ± 0.04 3.63 C = (0.065 ± D = (0.123 ± 0.008) • E 0 ' 8 0 9 ± °' 1 0.026) • E 0 - 6 2 1 * °' 1 (MeV) C(MeV) D(MeV) D/C R(%) 50 1.51 ± 0.05 1.82 ± 0.07 1.20 ± 0.06 6.66 130 3.31 ± 0.06 3.72 ± 0.07 1.12 ± 0.03 5.41 200 4.79 ± 0.15 4.88 ± 0.18 1.02 ± 0.05 4.84 300 6.84 ± 0.20 6.21 ± 0.23 0.91 ± 0.04 4.23 C = (0.055 ± D = (0.130 ± 0.007) • g0.843 ± 0.1 0.02) • g0 .685 ± 0.08 The upper table i s for a "pencil" beam, while the lower table i s for a diverging beam. - 57 -2) C = C 0 ( E Y ) 0 * 8 D - D ^ E y ) 0 * 6 where C0 and DQ were calculated from the f i t to the radiative capture peak at several different energies. For comparison, each of these relations were separately used to find the charge exchange cross-section at several different energies and angles, and the difference proved to be less than 1%. Also, because of excellent s t a t i s t i c s , the variation of the parameters C Q and DQ was found to be smooth and small enough so that the f u l l shape of the it 0 portion of the spectrum could be fi t t e d by a convolution of P(EY,A,B,C,D) for constant C 0 and D0 values with the energy distribution of n° (•* 2y)-III .2 .3 Cuts and the Production of Final Spectra The goal of this experiment was to measure differential and total cross sections for the reactions: 1) Tt" + p -• Tt° + n (n° •* 2y) 2) TC~ + p •* y + n for several pion energies . Reaction 1 produced a neutron and a n° which subsequently decays into 2 Y ~ r a y s a n d produces a photon energy spectrum which is smoothly distributed between 15 MeV to a couple of hundred MeV at our beam energies. The selection of acceptable events for f i n a l analysis was done by restricting the events to correspond to the arrival of pions in the beam time of flight spectrum, and to the selection of y-rays (and thus rejection of neutrons). This task was performed in the following steps: 1) Neutral cut:- Charged particles events In TINA were rejected using the signals in S4 which were identified by a logic signal in the coincidence - 58 -buffer. This cut guaranteed that a l l of the events are now related to neutral particles, I.e. y-rays and neutrons. Then, a l l of the histograms with this cut were printed, and the window for the pion beam time of flight was chosen. 2) Pion cut:- Together with cut 1, cuts were made on the time of flight of the pion down the beam l i n e . This cut removed the muons and electrons and thus the electron induced bremsstrahlung from the beam as well as some random background. The application of these cuts guaranteed that a l l of the events are now pion-induced. At this stage, a l l of the histograms were printed again and the window for y-ray time of fl i g h t was chosen. Fig. I l l .5 shows the typical time distribution of the beam particles (S1.S2.S3.) with respect to the r . f . buckets of the primary proton beam. The beam composition at S3 is obtained from this histogram. The pion's window is also shown. 3) y - r a y time of flight cut:- Cuts were made on the TINA time of flig h t spectrum to select y-rays and reject the neutrons. Together with cuts 1 and 2_, this cut ensured that almost a l l of the events are related to the y-rays which are generated either from the decay of u° or from the radiative capture reaction. This was one of the most delicate cuts that we made, and great care was taken in deciding the position of the cuts. For some of the time of flight spectra, a time walk has been observed. This has been corrected by using the scatter plot of time of flight vs . energy. Some low energy background y-rays and neutrons were also present in the energy spectrum. F i g . I l l .6 shows the time distribution of neutral - 5 9 -CM o CO h-Z O O 7 14 21 TIME (ARBITRARY UNITS) Figure III.5 The time distribution of beam particles (T^ - 110 MeV). - 60 -20, 15 O to 101 »-ZD O o TIME O F F L I G H T S3 - T I N A n '20 25 30 35 40 45 TIME ( A R B I T R A R Y UNITS) 50 Figure III.6 The tine distribution of neutral events in the TINA crystal (TL • 110 MeV). This histogram was used to separate gammas from neutrons by accepting only events within the window indicated. - 61 -events in TINA. This was used to separate y-rays from neutrons by accepting only events in the window indicated. The efficiency of the cuts was determined by selecting high energy signals in TINA which could be assigned unambiguously to the it + p y + n reaction and determining the loss of the events by applying the selected cuts to the data. In this way the cut efficiencies for pions in the beam and y-rays in Tina were found to be 99% and 98%, respectively. At each angle of the y-ray detector several runs were recorded with the target f u l l and empty. In the data analysis, after applying the f i n a l cuts, normalized empty target runs were subtracted from the corresponding f u l l target runs. This subtraction helped the further removal of the background generated by the sci n t i l l a t o r s and target walls and produced the f i n a l TINA differential pulse height spectra for f i t t i n g . A typical energy spectrum Is shown in Fig. I l l . 7 . At T^ = 50 MeV and 66 MeV, y-rays from stopped pions are present, in addition to the y-rays from the radiative capture and charge exchange reactions for in-flight pions. The clearest evidence comes from our earlier work at 27.4 MeV (SAL84) . The at-rest events can occur when a pion enters the target and scatters along the direction of the target plane, so that the effective target thickness becomes greater than the pion range. Although the number of such events is small, and almost n i l at higher energies, every pion that stops in the liquid hydrogen w i l l produce a y-ray or a u°, which subsequently decays into two y-rays. During the experiment, several runs were performed with stopped pions by introducing an aluminum degrader between S2 and S3. They Figure III .7 Typical y-ray energy spectra in TINA with a l l cuts applied and the f i t . Figure III .7 Typical y-'ay energy spectra in TINA with a l l cuts applied and the f i t . CHMNNC15 Figure III .7 Typical y-ray energy spectra in TINA with a l l cuts applied and the f i t . - 65 -have provided good spectra for y-rays resulting from stopped pions for different collimators. These have already been illustrated in Fig. I l l . 3 and F i g . I l l .4 for 15.24 cm and 25.4 cm collimators. I l l .3 Radiative Capture Data III .3.1 The Analysis of the Radiative Capture Part of the Spectra One of the major d i f f i c u l t i e s of this experiment is the small radiative capture cross section relative to the competing charge exchange reaction. Fortunately, the good energy resolution of TINA made i t easy to f i t a l i n e -shape to the capture peak and thus to obtain the area under the peak. In order to obtain the yield of the radiative capture y-rays, the detector response function was characterized by the form: y = A * exp((E-B)/D) * (1-erf[(E-B)/C]) + AA * (1-erf[(E-BB)/CC]) (2) That Is, the area of the y-peak was determined by f i t t i n g both the capture y-peak and the background from the charge exchange reaction. This background is characterized by the second part of the equation ( 2 ) . This is a seven parameters function and the parameters AA, BB and CC are related to the back-ground. Figs. I l l .8 show the spectra and the f i t (solid line) for several different energies and angles • The area under the capture peak can be determined from the equation: - 66 -The y-ray spectra and the f i t for (tt~p yn) reaction at several different energies and angles. The background for the n* decay ( T I ° • 2Y) is also shown. - 67 -The y-ray spectra and the f i t for (ii""p-*• yn) reaction at several different energies and angles. The background for the n° decay ( i t ° •»• 2y) is also shown. - 68 -A(A,C,D) - 2AD exp(C2/4D2) (3) where the parameters A, C and D are found from f i t t i n g . It should be emphasized that the parameters C and D used to f i t the Y - r a y s from (n + p •+ y + n) reaction in f l i g h t differed significantly from those used for stopped pions because of extra width due to f i n i t e target thickness and kinematic broadening. For a l l of the beam energies, the stopped pion components gave a nearly zero contribution to the radiative capture spectra. In the course of analysis, i t was noticed that for each energy there were one or two angles with differential cross sections not compatible with the pattern set by the rest of the angles. After scrutinizing the f i t t i n g parameters, the source of this erratic behaviour was found. It was observed that for these angles, the parameter B, which is approximately the peak position, was moved either to the extreme lef t or right side of the peak, i.e., in the t a i l region. This shift has in turn increased or decreased the value of the parameters C and D significantly. The area for the capture peak is linearly proportional to the parameter D (equation 3) and therefore any change in D affects the area directly, and hence the differential cross section. Further comparison revealed that, while for the rest of the angles the value of D/C was almost constant (typically between 1 and 1.1) at each energy, i t became much smaller or larger than this value for the peculiar cases. Therefore, in order to avoid any gross fluctuation, we found i t necessary to f i x the ratio (D/C). - 69 -After performing the f i t and finding the area under the capture peak, the di f f e r e n t i a l cross section at each angle was calculated using: N r da -, y 0 L AO J dQ J l a b N _N Q e (4) u p v ' where N is the area under the peak N 1 2 3 ( f u l l ) N = N- - . - — ; —r- N _ (5) Y0 f u l l N 1 23 (empty) empty v ' Relation (5) means that the number of .yQ, i . e . the area, was found after a normalized empty target run was subtracted from f u l l target run. is the number of protons in the target which is found from N = A P t . (6) p m cos 4> where t is the target thickness, p is the density of liquid hydrogen which i s equal to 0.0703 g/cm3, § is the angle of target with respect to beam, m = 1.0078 Is the atomic mass of hydrogen, and A = 6.022 * 10 2 3 is Avogadro's number. N _ is the number of pions which is found from it r N - = N 1 2 3 * f (E ) (7) - 70 -where N 123 is the total number of particles in the beam and f (E ) i s the pion Tf fraction of the beam and is a function of pion's energy, ft i s the solid angle of the detector and e is the efficiency of the detector, which is almost 100%. So, using equation (4), the differential cross sections i n the laboratory system were calculated at different angles and different energies for TT + p Y + n reaction. Figs. III.8 show the radiative capture peak, the background from charge exchange reaction, and the f i t (solid line) for several different energies and angles. III.3.2 Application of the Principal of Detailed Balance Most of the previous data came from the photoproduction of negative pions experiments, and almost a l l of the theoretical calculations were performed for this time reversed channel, i.e. ( Y + n •*• n + p). Therefore, for the sake of comparison, the ( T T + p •*• y + n) cross sections should be converted to the reverse process. To this purpose, we compare the photoproduction of negative pions with the time reverse process at the same energy in the center of mass system, using the principal of detailed balance (KAL64). This principal states that, for the reaction a + b •*• c + d, we can to Pimeson Photoproduction write: (8) do ( CM unpol * (2S c + D(2S d + 1) X (S.m2,, m2) - 71 -where AB denotes the transition from the i n i t i a l state |A> containing particles a and b to the f i n a l state |B> with particles c and d, and vice versa, is the spin of particle i , and the factors \ appearing on both sides of the equation (8) are proportional to the square of the momenta in the center-of- mass system. This principal can, without essential modifications, be applied to these two processes. The only change which is necessary is to remember that the s t a t i s t i c a l factor of the photon is two because this particle has only two independent polarization directions (no longitudinal polarization state), in spite of the fact that the spin i s 1. In this way we find \(S,M2,o) a(it + p > Y + n ) , = 2 2 a (y + n TC + p) (9) \(S,M ,m2 _) P TC where /s is the total energy in the center-of-mass system, i s the mass of particle i , and \ is defined in the following way: \(x,y,z) = x 2 + y 2 + z 2 - 2xy - 2xz - 2yz The center-of-mass cross section for the radiative capture is calculated from the following relation cm lab cm - 72 therefore, the cm. differential cross section for y + n + i r ~ + p reaction can be found from equation ( 9 ) . The angular distribution of photoproduction reaction in the cm. system could be expanded as a series of Legendre polynomials. The experimental data are fi t t e d to this equation. The constants A£ in equation (11) are parameters which describe the angular distribution and are quantities to be determined by the f i t . Since the effects of d-wave scattering are not important (less than 1%) at incident pion energies of less than 200 MeV, i t i s assumed that scattering occurs only in the s and p partial states and only the terms related to £ - 0, 1 and 2 are considered. We have also tried the so called Moravcsik f i t and the results were the same as the Legendre polynomial f i t . The total cross section i s calculated from n - I A £P £(cos 6) (11) CM. £-0 o (12) TOT The results for the angular distributions w i l l be discussed later in Chapter V. - 73 -III.4 The Analysis of the Charge Exchange Part of the Spectra The f i n a l spectra for the charge exchange part of the analysis were produced by subtracting the radiative capture peak from the rest of the spectra. The parameters from the f i t were used to construct the capture peak. This subtraction removed the contamination due to radiative capture's low energy t a i l , upon which the charge exchange spectrum is superimposed. To obtain the angular distribution for the charge exchange reaction i s more complex, because each angle setting of the Nal(Tl) spectrometer contains information about a l l of the neutral pion angular distribution. This is because neutral pions moving toward the detector, which decay with the resulting y-ray hitting the detector, w i l l be recorded as high energy events, whereas pions moving away from the detector w i l l produce low energy y-ray events. Therefore, the energy distribution of the y-rays is related to the angular distribution of the neutral pions. Thus, to determine the charge-exchange differential cross section, the y-ray energy spectrum from the decay of i t 0 has to be unfolded. The unfolding procedure, which is described by Kernan (KER60), is briefly summarized here. We assume an angular distribution for the i t 0 in the center of mass, calculate the r e l a t i v i s t i c transformation to give the energy and angular distribution of the y-ray in the laboratory system, and then fold this result with the empirical line shape of the spectrometer. If one uses a partial wave expansion, the angular distribution in the center of mass of charge exchange reaction can be expanded by a series of Legendre polynomials, - 7 4 -do-o(e') a -35 \mQWoa e'> ( 1 3 ) where Q" is the angle of emission of the it 0 in the center of mass system, measured relative to the incident u direction. The differential cross section for the emission of y-rays is obtained from this equation by application of the addition theorem for Legendre polynomials. This cross section can then be subjected to a Lorentz transformation to obtain the energy spectrum of y-rays emitted at an angle o in the laboratory system. So, the spectral intensity of the y-ray produced by neutral pion decay and detected in the laboratory can be shown to be (B0D54) N Q e I n I N , A \ . E y A p r c o s G ~ P i n K , a ; 8 0a 0k" (1 - 8 cos a) z = 0 1 A 1 " P c o s a * r 1 f, ; ^ }-i (14) PALp^" 1 " y 0yk ( 1 - 8 cos a ) J J where is the number of protons In the target traversed by an incident pion, Q is the solid angle of the detector, e Is the total efficiency, and I is the Tt~ intensity. The kinematic factors are the following: BQ and y 0 are the r e l a t i v i s t i c parameters of the u° in the cm. system, k is the energy of the y-ray in the laboratory system, k" is the energy of the y-ray in the rest frame of the i t 0 , 8 and y are the r e l a t i v i s t i c parameters of the cm., and a i s the laboratory angle of y-ray, i . e . the angle of observation. To f i t the - 75 -y-ray energy spectrum from the T C u decay, the spectra in equation (14) must have the spectrometer response function and detection efficiency folded into them. So, the expression for I(k,a) was folded with the detector response function P(E^,A,B,C,D) which has a typical 4% resolution. An added complication at low energies is the stopped pions in the target. Unfortunately, a stopped u ~ always gives off at least one y-ray, and hence a contamination in the i n - f l i g h t y-ray spectrum. As the characteristic spectrum for the stopped n~ is well established, i t turns out that this added complication does not seriously affect the f i t t i n g for the in-flight TC~ reaction. Therefore, to optimize the f i t , the stopped pion y-ray spectrum and a low energy exponential background, which is easily seen below 20 MeV, are also f i t t e d . It is to be noted that the constants, A^  , which appear explicitly in equation (14), are the parameters which describe the angular distribution of n° in the cm. system and are the quantities to be determined by experiment via f i t t i n g . Since the terms in I(k,oc) for A > 2 give a negligible contribution and the effect of d wave and higher partial wave scattering are not important (less than 1%) at incident pion energies less than 200 MeV, which is the case in this experiment, the upper limit of A was set to two. If, however, the d-wave scattering i s of interest, the number of Legendre polynomials must be increased to include P3 and P^  terms. The experimental spectra were fitted to the equation, 2 f(x) = P exp (-ax) + A'S + I = 0 (15) - 76 -where the exponential term i s related to low energy background, A'S i s related to the stopped pion spectrum, A Q, A^ and A 2 are the familiar Legendre polynomial coefficients in equation (14), and GQ, Gj, G 2 are the convoluted form of Pj, ? 2 a n c* ? 3 i n I(k,a)., Since the probability of having a stopped pion i s almost zero at higher beam energies, A' was set to zero for beam energies higher than 70 MeV. To summarize, the method consists of measuring the absolute number of y-rays and their energy distribution for several angles of observation a. The results are used to determine the parameters A^, which in turn specify the angular distribution of emitted n°. Figs. III.9 show the charge exchange spectra and the f i t for several different energies and angles. Since the errors of the A^ were angle dependent (see Appendix B), i.e. the sensitivity to in I(k,a) is a function of a, the fi n a l values of A^ and the corresponding errors for each beam energy were determined using a weighted average of individual angles. Assigned s t a t i s t i c a l errors correspond to a change in x2 P e r degree of freedom by one. The sensitivity of the fitted A^ was tested to uncertainties in the background contribution and the shape of the fi n i t e resolution function, in particular the parameters C and D. These uncertainties were always smaller than the s t a t i s t i c a l errors (less than 1% for A 0 and the values of and A 2 were hardly affected by these uncertainties) and were included in the fin a l error of A^. The sensitivity of the results to the multiple scattering in the target and the f i n i t e solid angle of the y-ray spectrometer were also investigated. Both effects proved to be negligible (less than 1%). Finally, the integrated value of differential cross section gives a - 4TCA0, - 77 -Figure III.9 The charge exchange y-ray spectra and the f i t at several different energies and angles. The contribution of different Legendre polynomial terms are also shown. - 78 -Figure III.9 The charge exchange y-ray spectra and the f i t at several different energies and angles. The contribution of different Legendre polynomial terms are also shown. - 79 -which is calculated from parameter AQ . It should be emphasized that, because of excellent s t a t i s t i c s , i t was possible to obtain results with small errors, and i t was possible to compare results at the various angles for consistency. I l l .5 The Renormalization of the 50 and 60 MeV Data The study of low energy pion-nucleon interaction for radiative capture and charge exchange reactions was carried out in two stages. The f i r s t stage had employed the low energy pion channel M13 at TRIUMF and investigated these reactions at T^ = 27.4 and 39.3 MeV. For the second stage, aside from a test run at M13, the higher energy pion channel, Mil, was used. The second phase consisted of three experiments: a) September 1981 run - This was a two week run and we collected data at T = 50 and 66 MeV; Tt b) February 1982 run - This run was also two weeks long, during which we repeated the collection of data for T^ = 50 and 66 MeV, and also managed to get some data at T =80 and 95 MeV; & it c) July 1983 run - This was our most successful run, and continued for more than a month, during which we collected complete data at T^ = 80, 95, 110, 125 MeV, repeated 50 and 66 MeV runs for four different angles, performed a test run at T^ = 45 MeV at two different angles, together with several stopped pion runs using six and ten inch collimators for calibration. After the September 81 experiment, a preliminary analysis of radiative capture data had indicated that the resulting differential cross sections were between 20% to 30% lower than what we had expected. Subsequent charge-exchange - 80 -analysis of this run supported our worries that we had a normalization problem. The only d i f f i c u l t y that we had during the September run was due to the hydrogen target, because we had some problems f i l l i n g the target and there were occasions that the target was only three quarters f u l l , or the hydrogen condensed in the buffer volume. Aside from this problem, we were not able to find any malfunctioning that could explain the normalization problem. After the September run, a change was made in the design of hydrogen target, i . e . we used a guard ring to stop bubbles traversing the target, and a s l i t was placed in the insulation so that we could monitor the H 2 l e v e l . Our February 1982 experiment was also plagued by a faulty TDC unit which caused neighbours in the camac crate to malfunction; one example was that the scalars overflowed. However, we managed to collect some data in the last few days of the run. Therefore, we approached our last run very cautiously, with a l l sources of earlier problems being monitored carefully. First we repeated the phase one experiment at T^ = 45 MeV to see whether we could reproduce the previous results. After this test, we repeated the 50 and 66 MeV runs at several different angles. The results were very encouraging because not only did we confirm our results at T^ = 45 MeV, but also the new results for 50 and 66 MeV were about 25% higher than the September results and supported our suspicion concerning a normalization problem. Encouraged by these results, we chose our next energy at T =95 MeV, so that we would be able to compare our charge - 81 -exchange total cross section at this energy with the accurate measurement of Bugg et a l . (BUG71). The agreement proved to be excellent. At this stage we became confident that we had eliminated the problem which had af f l i c t e d our earlier runs, even though i t was not clear which of the problems had been the cause of the incorrect normalization. Because we did not have enough time to repeat the 50 and 60 MeV runs for complete angular distributions, and the data were collected only for four different angles (two forward and two backward angles), we decided to renormalize our September data with respect to the July's results. The f i n a l analysis of these data showed that there was just a single factor (about 22.5%) involved In the renormalization of different angles, both for the charge exchange and the radiative capture reactions. For the f i n a l results, at 50 and 60 MeV, we have therefore combined the results by using the differential cross sections for those angles available from the July run and then supplementing these with the renormalized data from the September run. A l l other data have been taken from the f i n a l run only. - 82 -CHAPTER IV The Theory of Pion Interactions with Nucleons IV .1 Introduction In this chapter, some formulae for Tt-N scattering w i l l be introduced. It is standard practice at low energy to express the observables as functions of parameters called phase shifts, that i s , the shift in phase of the wave function at a large distance from the scattering center, resulting from the scattering process. There w i l l be a separate phase shift for scattering i n each state of total angular momentum, isospin, and parity. IV .2 On energy-shell amplitudes; partial waves Scattering reactions are described in terms of probability amplitudes relating the i n i t i a l and f i n a l asymptotic states of the combined system of projectile and target. If we denote these states by a and b respectively, we may define two matrix amplitudes: the S-matrix S^a and the transition or T-matrix <b|T|a>, which are related as follows: Sba = 5 b a ~ 2 1 1 1 5 < Eb " V < b l T l a > The labels a and b stand for the quantum numbers of the states asymptotically, that i s , at such a large distance where the projectile and target no longer Interact. Then, for a meson plus nucleon, we may write - 83 -|a> - (2n)- 3 |t,p,o> (2) where k and p are the linear momentum of the meson and the nucleon respectively, and a stands for the quantum numbers for spin, isospin (and i f necessary, strangeness, etc.)* For Tt-N system, a specifies the spin projection mg of the nucleon and the isospin states t 3 (pion) and T 3 (nucleon). The S-matrix i n (1) i s defined only for the f i n a l state energy E. b equal to the i n i t i a l state energy E , where these energies are the sum of the internal and kinetic energies of projectile and target. The S-matrix i s referred to as being "on-energy-shell" with E = E a = E^. The S-matrix i s unitary, cb ca ba T S*v S = 6 V (3) C where the sum is over a l l possible asymptotic states (channels) with energy E = E. = E = E. The relation given by (3) expresses the conservation of c b a probability (flux) in scattering processes. The T-matrix in equation (1) i s also "on-energy-shell" as well. For an isolated system the total momentum i s conserved, so we may write <b|T|a> = T b a 6 (\ - P a) ( 4 ) where P = K + P. The cross section to some particular set of fi n a l states a b, for a given i n i t i a l state a, is given by - 84 -a b where V is the i n i t i a l relative velocity of the projectile and target, and the sum or integral b is defined by the choice of f i n a l states measured. It is convenient to work in the center of mass (cm.) frame ()? = f, = 0 ) , with a b i n i t i a l and f i n a l states given in (2) specified by the meson momentum K, K', respectively: |a> = ( 2 u ) " 3 / 2 |K, -k, a> |b> = ( 2 r c ) - 3 / 2 |K\ -K\ B>. ( 6 ) We define the elastic scattering amplitude as * £ p a = (-2u)2 m \ a - - «- < K', P|T|K,a> (7) * where m is the reduced mass for the meson and nucleon: m = [e (K) + e ( K ) - 1 ] - 1 . The differential cross section for elastic 1 m u 1 scattering (unpolarized) is given by spins where the sum is over the spin projections of the nucleon in the i n i t i a l and f i n a l states. The amplitude given in (7) and the cross section given in (8) - 85 -depend on the energy E = E «• E, , the directions K, K', and (before summing) £1 D the nucleon spin projections. If we introduce the Pauli spin matrices a for the nucleon, we may represent the amplitude in (7) as a spin matrix. Invariance under rotations in space and parity inversions limits the form of f 0 to B a f p a(E,K,K') = f(E, 6 ) + g(E , 9)ia.n n (9) (K x "k') \t x£'| where 6 is the angle between K and K'. The differential cross section given In (9) can now be written as ^ ( E , 6 ) = | f ( E , 9 ) | 2 + | g ( E , 9 ) | 2 ( 1 0 ) since f ( E , 9 ) has the same spin dependence as the i n i t i a l state, i t is called the spin-independent (or central) amplitude. On the other hand, g(E,9)a«n depends on the spin and i t is called the spin-dependent amplitude. The amplitude in (9) can be expanded in partial waves by expanding the f and g separately: f ( E , 9 ) = I {(A+l)f (E)-Wf (E)}P (cos 9 ) ( 1 1 a ) A = 0 - 86 -00 g(E,9) = I {f A +(E)-f A_(E)}P^(cos 6) sin 9 ( l i b ) A - 0 in terms of the Legendre functions P^(x), x - cos 6, and their derivatives P'^(x) = ^ " ^ ( x ) * The partial-wave amplitudes f^ +(E) are defined for orbital angular momentum A and total angular momentum j = A±l/2 of the meson-nucleon system in the cm. Due to invariance under rotations and parity, these are both conserved quantum numbers. This means that the S-matrix in (1) is also diagonal in A and j . Using the unitary condition given by (3) and the definitions in (1), (4), (9) and (11), one finds that the f A + ( E ) can be expressed in terms of two real quantities, the phase shift 6^ +(E), and the inelasticity parameter TK (E), in the form The parameter n measures the fraction of flux lost from the elastic channel for each A and j ; n = 1 implies no inelasticity, i . e . a l l flux conserved In the elastic channel, and 0 < n < 1 otherwise. For purely elastic scattering, equation (12a) takes the usual form (n 9 +(E) exp 2i6 9 + ( E ) - l ) ) 2iK (12a) exp[i6 0.(E)]sin 6,.(E) K (12b) - 8 7 -In the limit of very low energy, the partial-wave amplitudes f^ +(E) a K For A = 0 and 1 i t is conventional to introduce the scattering lengths and scattering volumes: a = J^ Q f 0+(E) s-wave (13a) a, ° v_. n p-wave (13b) J=l ± 1/2 K 0 K2 These quantities may be complex i f there is inelasticity at zero energy (eg., K~p •*• TC°A); but, for the up case, the inelasticity due to tc~p -*• ny is usually neglected so they are taken as real. IV.3.1 Hadronic isospin, symmetry Heisenberg, who introduced the concept of isospin (HEI32), suggested that, as far as the purely nuclear interactions were concerned (electro-magnetism being "turned o f f " ) , protons and neutrons behaved so similarly that they could be regarded as two states of the same particle, the nucleon. The strong nuclear force was regarded as "blind" with respect to the charge label: an arbitrary linear combination of proton and neutron wave-function was then just as good as a single proton or neutron wave-function. Thus redefinitions of proton and neutron wave-function could be allowed, of the form 6 •* a>' = a 4> + 8 4 > P P P n (14a) - 88 -(i) 6' = v <N + 6 <1> n n p n (14b) f o r complex constants a, 8, y, 6. Equations (14a) and (14b) can be compactly w r i t t e n i n terms of a t r a n s f o r m a t i o n law f o r the two component nucleon i s o s p i n o r where U i s a u n i t a r y complex 2 x 2 m a t r i x . Heisenberg's proposal, then, was that the physics of strong i n t e r a c t i o n remained the same under the transformation (15b): i n other words, a symmetry was i n v o l v e d . I t should be emphasized that such a symmetry can only be exact i n the absence of electromagnetic i n t e r a c t i o n s and w i t h the strong i n t e r a c t i o n being i n v a r i a n t under SU(2). Both are now known to be sources of s l i g h t nonconservation, so i s o s p i n i s an i n t r i n s i c a l l y approximate symmetry. To preserve the n o r m a l i z a t i o n , I t i s r e q u i r e d f o r U to be u n i t a r y : where 0 i s a r e a l number. We can separate o f f such an o v e r a l l phase f a c t o r from the transformation mixing p and n. This i s because t h i s corresponds to r o t a t i o n of the phase of both proton and neutron wave-functions by the same amount. (15a) namely, a, 4,' = u 4> (15b) UU+ = TJ+U = 1, det(UU +) = 1 and detU = e x p ( i e ) - 89 -IV.3.2 Isospin and the pion-nucleon system To describe the isospin states of a pion and a nucleon, a total isospin can be defined which is the vector sum of the isospins of the two particles. This means that there are two possible total isospins for the n-N system, I = 1/2, and I = 3/2. In general a given n-N state, it +-neutron for example, w i l l not be a single Isospin state, but w i l l instead be a linear combination of I = 1/2 and I = 3/2. The coefficients w i l l be just the Clebch-Gordan coefficients for the addition of angular momentum. Table IV .1 gives the u-N state functions in terms of the isospin state functions |l,I 3>. The hypothesis of isospin invariance means that the n-N interaction i s independent of I 3 . That i s , i t i s independent of the orientation of the isospin vector in isospin space. Since isospin space is assumed to be isotropic with respect to strong interactions, then like angular momentum in a central f i e l d , the total isospin is conserved; because in quantum mechanics as well as in classical physics the equations of motion can be solved for at most two bodies, elastic scattering w i l l play a central role in this subject. There are ten possible pion-nucleon reactions, viz.: (i) ( i i ) ( i i i ) (iv) (v) (vi) i r p •* n T p u ~ p • » • n ~ p u - p •*• i t ° n n + n i i + n i t + n •*• TC°P T t ~ n •*• n _ n - 9 0 r TABLE IV .1 The n-Nucleon State Functions |nN> i n Terms of Isotopic Spin State Functions |l,I 3> h+p> - IT . Y > |rc+n> - 4 | 4 . i > + 4 I 7 » 7 > |*°P>-4l 7 . 7 > " 4 l 7 ' 7 > |n°n> - 4 | - 7 > +4 | \ , - \ > |u-n> = ly , - 4 > - 91 -it°p Tt°p ( v i i ) n°n Tt°n ( v i i i ) Tt°n •*• Ti~p ( i x ) n°p •*• T t +n (x) Reactions ( i x ) and (x) are r e l a t e d to ( i i i ) and (v) by the p r i n c i p l e of time r e v e r s a l i n v a r i a n c e . The cross s e c t i o n s f o r a l l of the above r e a c t i o n s w i l l be equal to the square of some matri x element, a = |<TIN]M|TIN|>|2 (16) By the hypothesis of charge independence I i s conserved and the matrix element i s independent of I 3 . Thus, a l l of the above r e a c t i o n s depend only on two amplitudes: <l/2|M|l/2> = F x (17a) and <3/2|M|3/2> - F 3 (17b) where M i s a s c a t t e r i n g o p e r a t o r . Thus F . defined i n (12b) can be w r i t t e n 3 f o r each of the ten r e a c t i o n s as fJL± = YlfA± ( I = 1 / 2 ) + Y 3 f j t ± C1 " 3/ 2) (18) - 92 -The coefficients Yi a n d Y3 c a n D e simply derived from Table IV.1 and equation (17). They are given in Table IV.2. Differential cross sections for elastic and charge exchange pion-nucleon scattering provide most of the information we have about the on-energy-shell amplitudes f^^ or T^a in equation (7). Because it 0 beams are impractical and neutron targets are problematical, unfortunately only processes ( i ) , ( i i ) and ( i i i ) are readily accessible, so In practice there is l i t t l e redundancy. Spin polarization experiments are also necessary to separate the f and g parts of equation (9) for both isospin amplitudes. The requirement of isospin invariance leads to triangle relations (equation + 1.2) which connect differential cross sections of (it p) scattering and u~p charge exchange scattering (°" c x)' The experimental data must satisfy these inequalities i f isospin invariance is valid . IV .4.1. Phase shift analysis As mentioned before, for each state of total angular momentum, parity, and isospin there is a phase shift which gives a measure of the scattering in that state. Up to about 2 to 3 GeV the number of required partial waves is sufficiently small to make this a practical description of these scatterings. At higher energies, i t is inappropriate to use this approach. IV .4 .2 Determination of phase shifts A l l methods of determining the phase shifts involve the methods of 'least squares' in some way. With the assumption of isospin invariance, the phase shifts determine uniquely the following quantities: - 9 3 -TABLE IV.2 Isospin Scattering Amplitude Coefficients for the n-N Reaction fA± • nfA± C1 " 1/ 2> + Y3 fju; (I - 3/2) Reaction Yl Y3 H + p •• n + p 0 1 Tt~p * TC~p 2/3 1/3 i t ~ p •+• it°n - /2/3 /2/3 u +n -*• Tt +n 2/3 1/3 Tt +n •*• n°p - /2/3 • 2/3 n~n •+• Tt~n 0 1 H ° P * 7t°p 1/3 2/3 u°n + it°n 1/3 2/3 it°n •* n~p - /2/3 • 2/3 n°p -»• n +n - /2/3 •2/3 - 94 -1) the TC p total cross sections a"1" _ and o*~ ' r tot tot 2) the real part of the elastic forward scattering amplitude, Ref +, Ref" 3) TC p inelastic cross sections cv\ , a~. ' in i n + 4) TC p and charge exchange elastic differential cross sections (unpolarized _ N da + da - da 0 targets) _ , _ , _ + 5) polarization of the recoil proton in TC p elastic scattering and of the recoil neutron in charge exchange (unpolarized targets) ? +, ?~, ?° ± J . n 6) rotation parameters in TC p and charge exchange scattering A , A , A u, R+, R-, R° . A l l of the above parameters (quantities) can be measured experimentally except Ref which can be determined from dispersion theory. A % c a n thus be defined + between a l l the quantities that are known experimentally (and Ref ) and the predictions from the phase s h i f t s . There are two techniques for determining the energy variation of the phase s h i f t s . In one method the phase shifts are determined independently at each energy. They are f i r s t determined at the lowest energy for which data is available. Then these phase shifts are used as i n i t i a l values for the least square analysis at the next higher energy, or the phase shifts can be extrapolated to the next higher energy and these values used as input at the higher energy. In this way a set of phase shifts is obtained at various discrete energies. Because invariably data is used from more than one experiment, each with i t s own systematic errors, small phase shifts tend to - 95 -have a somewhat erratic behaviour. This method is called energy-independent analysis. In the second method the phase shifts are determined at a l l energies simultaneously. The phase shifts are parmeterized as a function of energy and a x2 is defined which includes the data at a l l energies. This method is called the 'energy-dependent analysis'. To describe the energy dependence i t is normal to have resonant and non-resonant features, each with their own parameters. As long as there is more than one solution, additional experiments must be performed. Generally more information is obtained by doing different kinds of experiments rather than adding accuracy to data already obtained. Because of the various ambiguities, a bare minimum require-ment is to have data on the total cross section, differential cross section, polarization and Ref from dispersion theory. IV.5 Optical theorem and total cross-sections The optical theorem, which follows from the unitary relation (3), relates Imf (E,0°) to the total cross section: ~ Imf (E,0°) = o t o t ( E ) (19) where °tot ( E ) = ° e l ( E ) + ° i n ( E ) - ( 2 0 ) This theorem can be used to derive an important relation between the n +p total cross section (a+) and n~p total cross section ( a ~ ) : - 96 -a+ = ^Im< 7 l+p|M0| 1 [+p> (21) a~ = Im < TC~P|M° |it~p> where M° is the scattering matrix at zero degrees, writing the scattering amplitude at zero degrees as A 0(1/2) and A 0(3/2) for I = 1/2 and I = 3/2 states, 4 = I T l m A ° ( 3 / 2 ) . (22) the r e f o r e o- = 1/3 a+ + | a T ( 1 / 2 ) . (23) The Ref (E,0°) i n theory can be obtained from the a n a l y s i s of i n t e r f e r e n c e of the Coulomb and strong i n t e r a c t i o n s c a t t e r i n g , at small angles, e x t r a p o l a t i n g to 0 = 0 ° . The r e a l and imaginary parts of f(E,6) are r e l a t e d through forward d i s p e r s i o n r e l a t i o n s , of which there have been many a p p l i c a t i o n s to nN s c a t t e r i n g (HAM63, BRA73). IV .6 Pion-nucleon dynamics, Born terms The pions are represented by a f i e l d operator $(x) which i s pseudoscalar i n space and i s o v e c t o r i n i s o s p i n space. The f i e l d operator can be w r i t t e n i n terms of a n n i h i l a t i o n and c r e a t i o n operators f o r plane wave s t a t e s : - 97 -<pa(x) = / dkN k{a a(k)e^ , X + a+ (k)e _ i^ , X} (24) where a = 1, 2, 3 represent the "cartesian" components in isospin space. The isospin can be written in "spherical vector" form to correspond to the three charge states TC+, TC° , TC-. Using index V = 1, 0, -1, then a+ creates a pion of momentum tt and charge v where a+_ (£) = - / - j [a+ (tt) + i a+ (*)] a+ (£) = a+ (£) (25) = A y [a+ (tt) - i a+ (£)] and the operator a^ (tt) • ( - l ) V ( a + v ) + annihilates the pion of momentum tt and charge v. The interaction of pions with a nucleon is assumed to be linear in the meson f i e l d , and local in analogy with electrodynamics. It can be written in terms of a Hamiltonian density H(x): Hj- JdxH(x), H(x) ="!}(x) • l(x) (26) where ^j(x) is the nucleon current operator at the position x. Since $(x) is - 98 -pseudoscalar and isovector, ^j(x) must also be pseudoscalar and isovector, to give a scalar, isoscalar interaction . Therefore ^j(x) also carries an isovector index a: J a(*) a n (* t n e dot * n (26) indicates a scalar product in isospin space. The simplest nonrelativistic current operator with these properties is of the form j (x) = UkU x a • V (27) a m a Tt where the gradient acts on the f i e l d 4>a(x) only, and, t , a are the nucleon isospin and spin operators; f Q Is the dimensionless coupling constant. If we consider a DIrac nucleon, the simplest operators take the forms j a(x) = i g 0 T A Y 5 (PS) (28a) j a ( * } = & T a Y 5 Y * 7 ( P V ) ( 2 8 b ) where the four-gradient acts on the f i e l d 4>a • Tbe terms pseudoscalar (PS) and pseudovector (PV) are used to denote the two forms of coupling to the nucleon given in (28), although both operators are actually pseudoscalar, and combine with the PS pion f i e l d <J>(x) to give a scalar interaction in (26). The r e l a t i v i s t i c forms In (28) have proper Lorentz properties of a current. The absorption or emission of a pion by a nucleon are induced by the interaction given by (26). The matrix element of H between plane wave states for - 99 -absorption of a meson by a nucleon (TC + N •*• N) can be written as: <P'B|H I|P >a,kV> = /dx N^'^Hl J (x)|?a> (29) = (2TC)3 N 6(P'-P-t) <I'B|J (o)|Pa> K p, for nucleon momentum P, isospin a, 6 in the i n i t i a l and f i n a l states, respectively, and a pion of momentum t and charge p.. For emission, one can take the hermitian conjugate matrix element of (29). There are two interpretations for the matrix element in (29). From the point of view of perturbation theory, the plane wave states are unperturbed states of the pion and nucleon with equation (26) as the perturbing interaction. The particles have unrenormalized masses not corresponding to free physical particles. For the static current operator in (27) we can write <P'B|j (o)|Pa> -/toT|»-<B \o't x |a> with similar evaluation for r e l a t i v i s t i c cases given in (28). The second method of treating the transition amplitude is to use plane wave states of physical pions and nucleons, which propagate freely but with their proper masses . This can be characterized in terms of a new renormalized coupling - 100 -constant and a vert e x f u n c t i o n or form f a c t o r . For the s t a t i c c urrent operator, we ob t a i n <P'P| j (o)|Pa> = - /4~r7 J-<p |O'*T |a > F(£) (30) ^ it ^ where f i s the renormalized c o u p l i n g constant and F(ic) i s the form f a c t o r , which expresses the p r o b a b i l i t y that a p h y s i c a l nucleon w i l l remain a nucleon ( i n i t s ground s t a t e ) a f t e r absorbing a pion of momentum £. Consequently |l-F(1e)| i s a measure of e l a s t i c i t y , i . e . the p o s s i b i l i t y of reaching e x c i t e d s t a t e s of the nucleon. For the r e l a t i v i s t i c case, the matrix element i s w r i t t e n i n terms of a vert e x f u n c t i o n T: <P ' B | j ( i(o)|Pa> = g r p a (P',P) In p a r t i c u l a r , f o r the (PS) case, <P«6| j (o)|Pa> * i g <P'B|T^Y5|PGC> where: ( P ' 2 = P 2 = M 2, k 2 = m 2) In p e r t u r b a t i o n theory, the lowest' order TCN s c a t t e r i n g process induced by H - 101 -is analogous to Compton scattering of photons by charged particles. There are two terms, shown in Fig. IV.1, referred to as Born terms a) direct, and b) crossed. Figure IV.1 Diagram for Born terms: (a) direct; (b) crossed. In the first term, absorption of the pion precedes emission; in the second, the order is reversed. For a nonrelativistic nucleon, the Born t-matrix can be written as <k«,P\B| T(Ea)|t,$,a> - 6(k'+P'-k-P)(21t)3{<^ 1B|Ta((^ )+Tb(a^ )|^ a>} where Efl « M + where the direct (subscript a) and crossed (subscript b) terms are separated. In second order, using (30) with F -»• 1: <J'B|Ta(a^ )|^ a> - A * f 2 ( a . t ) ( a . K ' ) x T a (31a) <£'B|Tb(a^)|ka> ** f 2 (o*t)(o«t' )^i (31b) (-w, ^nr/Au). w, , p  v k ' it k k' These amplitudes may be decomposed into isospin and angular momentum channels and the partial wave amplitudes are given by (EIS80) - 102 -f 2 k 2 f2I,2j ( w k ) = " 3m*u. C2I, 2j (32) TZ K. where C = C 8 - C b: Cll = 8, C 1 3 = C 3 1 = 2, C 3 3 = -4. The r e l a t i v i s t i c calculation of the Born terms, based on a covariant formulation and the r e l a t i v i s t i c currents (28) proceeds on somewhat different lines and is explained in (BJ064). There are few differences at low energy for the p-wave terms: aside from kinematic corrections for the recoil of the nucleon the same result as (32) is obtained. The r e l a t i v i s t i c Born terms also include s-wave (and A > 1) TtN scattering, unlike the static model which gives only p-wave scattering. The forward-scattering Born amplitudes have singularities as a function of to^. This singularity is a pole at co^ = 0 in the static case, m2* which becomes two poles at tu^ = ± i"^") * n t n e r e l a t i v i s t i c frame. The value of the amplitudes at the pole defines the value of f 2 or g 2 which are related 2 m by f 2 =|^- (^M")2, ^ n e c o n t r i b u t i o n °* t n e pole term is extracted from the + evaluation of forward dispersion relations for it p scattering. A recent analysis of experimental data (BUG73) gives f 2 = 0.079 ± 0.001 (33) How do the Born amplitudes for p-wave scattering in the static approximation compare with the experimentally determined partial wave amplitudes? The Born approximation should be best at low energy: i t w i l l certainly not produce - 103 -resonant behaviour and w i l l not even remain unitary at high energy. It i s interesting to compare the scattering amplitudes (13b) (static approximation) 1 f2 a 2 I , 2 j = " 3 m^" °2T,2j ( 3 4 ) w i t h the experimental r e s u l t s i n Table (4.3) one f i n d s that the signs of equation (34) and the rough order-of-magnitude set by (33) are i n agreement wi t h the tabulated v a l u e s , but the r e l a t i v e s i z e s of the d i f f e r e n t a. - are z l fZj wrong. For p r e d i c t i o n s of the s-vaves, one must use the r e l a t i v i s t i c Born terms, and one f i n d s that the Born amplitudes agree poorly w i t h the s c a t t e r i n g length of Table IV .3, and that the p r e d i c t i o n s f o r (PS) and (PV) d i f f e r i n magnitude, although they lead to the same Born amplitudes f o r p-wave. C l e a r l y the Born terms alone do not provide a theory of nN s c a t t e r i n g . IV .7 The theory of pion photoproduction While pion photoproduction has been discussed g e n e r a l l y i n many a r t i c l e s (DON78, LAG81), the low energy r e g i o n (Ey < 450 MeV) o f f e r s a d d i t i o n a l features of p a r t i c u l a r i n t e r e s t . Over t h i s r e gion the Watson theorem i s v a l i d to a good approximation f o r a l l m u l t i p o l e s , so that a more thorough phenomen-o l o g i c a l a n a l y s i s i s p o s s i b l e than at higher energies. There are a l s o fewer m u l t i p o l e s c o n t r i b u t i n g s i g n i f i c a n t l y to the r e a c t i o n . At low energies there are three important diagrams which are c a l l e d the Born terms and are shown i n F i g . IV .2 ( a , b , c ) . The f i r s t two diagrams are c a l l e d the shake-off terms: (a) - 104 -TABLE IV.3 Pion-Nucleon Scattering Length and Volume s-wave scattering length &21 (in lO^m; 1) al 170 ± 4 181 ±8 a3 -92 ± 2 -89 ±5 (BUG73) (SAM72) p-wave scattering volumes &21 2J ^ n lO - 3™^ 3) l33 l13 '31 204 ± 5 -27 ± 6 -43 ± 7 -84 ± 10 (SAM72) 204 ± 16 -16 ± 5 -37 ± 4 -54 ± 8 (SAL74) - 105 -is the direct term and (b) the crossed term; the third term is called the photo-electric effect and occurs for charged pions only, because the pion has no magnetic moment and interaction occurs via the electric charge only. At higher energies many other diagrams become important, especially those including the A and higher resonances. From the shape of the total cross section It is clear that resonances dominate at most energies. The theory must therefore be extended to include these possibilities which are shown i n Fig . IV .3. In general the intermediate state can be any resonance [A(1236), N(1520), etc.]. Table IV .4 gives the lowest energy resonances in pion-nucleon system. Qualitatively, the u° reactions (y + p •* it 0 + p); (y + n •+ it 0 + n) are completely resonance dominated, while the % reactions (y + p •*• •KR + n) and (y + n + n~ + p) have, in addition, a large background, predominantly in the s-wave multipole, coming mainly from the photoelectric term. The Born terms themselves dominate at the lowest energies (Ey < 200 MeV) in charged pion production in agreement with the threshold theorems . The resonance is of particular interest because of the dynamics of i t s excitation and because i t provides an excellent means of testing the fundamental isospin and T-invariance properties of the electromagnetic current. In discussions of the low energy region, the partial wave decomposition has almost invariably been carried out in terms of the electric and magnetic multipole amplitudes and M +^ , which have a simpler threshold behaviour than the partial wave heli c i t y amplitudes. In terms of these amplitudes, the Watson theorem, assuming unitarity, T-invariance, and neglecting inelastic production, takes the form - 106 -(a) <t>) V" " V i A. . A y ~ -* (O Figure IV.2 Born terms for single pion production. (b) (e) Figure IV.3 Isobar and heavy meson terms for single pion production. - 107 -TABLE IV.4 Lowest Energy Resonances i n Pion-Nucleon System I « - 1/2 I - 3/2 P^MeV/c) Mass (MeV) L, J T(MeV) Pll(MeV/c) Mass (MeV) L, J T(MeV) 610 N(1440) 1,1/2 200 300 A(1232) 1,3/2 115 740 N(1520) 2,3/2 125 870 A(1600) 1,3/2 250 760 N(1535) 0,1/2 150 910 A(1620) 0,1/2 140 960 N(1650) 0,1/2 150 1050 A(1700) 2,3/2 250 1010 N(1675) 2,5/2 155 1440 A(1900) 0,1/2 150 1010 N(1680) 3,5/2 125 1050 N(1700) 2,3/2 100 - 108 -I I 16* where 6*+ is the TC-N phase shift corresponding to angular momentum j = £±1/2 and isospin 21. IV .7.1 The helicity,multipole amplitudes, and multipole analysis For pion photoproduction from a nucleon: y + N •*• TC + N the scattering amplitude is given by (BRA73) 1 CO, x3 >\*K1 >x2 2 q a q b a 1 < \ 3 , X j S 3 - l|X l f\ 2> e 1 ^ " ^ ^ ^ (9) where X = \^ - \2» P = ^3 ~ \ » ^1 » ^2 • ^ 3 a n c* \ a r e t n e f e l i c i t i e s of incoming and outgoing particles and functions d 3 are defined in Appendix A. A., p. If particle 1 is the nucleon and 2 is the photon, then \ 2 = a R d ^ = ±1/2, so that the i n i t i a l state can be labelled uniquely by X. = \± - \ 2 . The f i n a l state can be labelled by the h e l i c i t y of the nucleon u = ±l / 2« The i n t r i n s i c - 109 -parities of photon and the meson are each -1, and the spins are s^ = s1+ = 1/2, s 2 = 1, and s 3 =0, so that: <-u|SJ|-\> = - <u|S;j|\>. The matrix elements! : — / ^ {<u|S |X> - 6 } u,X 2 i / 9 a q b a are of the form given in Table IV.5. An alternative description of photo-production is given by expanding the incoming photon wave into eigenstates of the total angular momentum of the photon, L. These states may be divided into those with parity ( - l ) ^ + \ which are the magnetic multipole states ML, and those with parity (-1)^, which are the electric multipole states EL, and as the photon has spin 1, the minimum value of L is 1. The possible transitions from the electric and magnetic multipole states conserving both j and parity are shown in Table IV.6. For the f i n a l (meson + nucleon) state, the notation s, p, d is used for JI = 0,1,2 - partial waves, and the subscript indicates the value of j = £±1/2. The definition of the states |j,m,ML> and |j,m,EL> with a conventional normalization i s : |j,m;ML> = ^  |j,m; L; X" = L> /L /L+l and |j,m; EL> = 1 [-z^zz | j,m;L;A' = L+l> + ~ | j,m,L,A' = L-l>] /2L+1 /L+l /L - 110 -TABLE IV.5 The Matrix Elements , 3/2 1/2 -1/2 -3/2 1/2 A 3 -1/2 -cJ - I l l -TABLE IV.6 The Contribution of the Multipoles EL and ML to Photoproduction Photon State Total Angular Momentum j Parity Final it—N State El 1/2 - 8l/2 El 3/2 - d3/2 Ml 1/2 + Pl/2 Ml 3/2 + P3/2 E2 3/2 + P3/2 E2 5/2 + f5/2 M2 3/2 - d3/2 M2 5/2 - d5/2 E3 5/2 - d5/2 E3 7/2 - 87/2 M3 5/2 + f5/2 M3 7/2 + f7/2 - 112 -where A" is the photon's orbital angular momentum. For the f i n a l it-N state, the h e l i c i t y states |j,m,\> can be expressed in terms of states |j,m,A> i n which the orbital angular momentum A is diagonal and the matrix elements can be written (BRA72) as: A + 1/2 _ JI + 1/2 /A(A+2) r F _ -, f1 / 2 > 3 / 2 T z 1 — 1 E A + V Ea+i>- Mu+D-] (35a) ^ l / ^ l / Z = B " + = ^ t (*+2>(V+l)- " " + E(A+1)J3 <35b> ^l/V-l/2 = A" + = = W V+ " V + D J + U + 2 ) * ( M U + 1 ) - + E A + ^ < 3 5 c > ^ / i ? - ^ - D * + * , = / A < * + 2 t MA + " EA + " V + D " M(A+l)- ] < 3 5 d> In these equations the magnetic and electric multipole transition amplitudes , are labelled by orbital angular momentum of the fi n a l state A, and the ± signs refer to j = A±l/2. The angular distribution in the center of mass system for an unpolarized beam and target is given by: da q u) j . . * " I q H ^ Ii] <2^ + 1 ) f l / 2 , Xdl 1/2 ( 9> - 113 -The helicity amplitudes f^/2 \ a r e r e ^ a t e < ^ t o t n e multipole amplitude by equation (35). It is expected that a multipole amplitude E^ +, M^+, leading to a f i n a l state b_ with orbital angular momentum A, w i l l be proportional to q^ near threshold; and for low energies only multipoles with small values of A w i l l be important. Retaining the terms with A=0 and A=l, only, and using the expressions for d 3 given in Appendix A, we find that: f l / 2 3 / 2 ( Q A ) = " ~= ( E1+ " M l + ) S i n 9 C O S ( 9/ 2> e l* f 1/2 1/2 ( 9 , < 1 > ) = ~= t 2 ( M l - + E 0 - ) + ( M1+ + 3 E l + ) ( - 1 + 3 c o s 9 ) 1 c o s 9 / 2 ' /2 f 1 / 2 )_ 1 / 2(e,<t>) = ~ [2(MX_ - E ^ ) - (M 1 + + 3E 1 +)(1 + 3 cos 6)]sin 9/2 e" 1* fl/2,-3/2 ( 9' < t > ) = ~ | ( E1+ " M i + > s i n 9 6 i t l ( e / 2 ) e " 2 i ( , , (36) Examining terms for large values of A, one can see that the partial cross section arising from a single multipole amplitude has the property that the angular distribution depends only on j and L, i.e . the total angular momentum and multipole order. For example, both dipole terms and M lead to a constant angular distribution, and the dipoles with j = 3/2, , and to - 114 -a distribution proportional to (2 + 3 s i n 2 9 ) . The properties associated with multipole analysis of single pion photoproduction is given in Table IV .7. At energies close to threshold, the cross section is observed to be isotropic, and an s-wave f i n a l state is expected, the transition must be due to electric dipole E , while around the A (1236) resonance with spin and parity 3/2+, the 0+ most important transition is the The experiments may be analyzed in terms of the multipole amplitudes in much the same way as phase shift analyses for -rtN scattering are carried out, and the ambiguities that arise may be eliminated in a similar way by measurements of the polarization of the recoil nucleon or by experiments with polarized photons. The number of parameters required at each energy can be reduced by relating the phase of each photo-production amplitude to the itN phase shift in the f i n a l state. IV.7.2 The electromagnetic interaction and isospin The electromagnetic interaction does not conserve isospin, but neverthe-less behaves in a well defined manner under an isospin transformation. As photoproduction has a small coupling constant, theory can be used to show that the electromagnetic current operator must transform with respect to rotation in isospin space like a mixture of an isoscalar (1=0) and an isovector (1=1). The i n i t i a l and f i n a l values of the third component of the isotopic spin must be identical in order to conserve charge. The physical amplitudes are as follows: - 115 -TABLE IV .7 Multipole Analysis of Single Pion Photoproduction Photon State J P K TC Transition Amplitude Energy Dependence Angular Distribution Ml 1/2+ 1 M l " p3 1 Ml 3/2+ 1 M1 + p3 2 + 3 s i n 2 e E l 1/2" 0 Eo+ P 1 E l 3/2" 2 E 2 " P5 2 + 3 s i n 2 e E 2 3/2+ 1 E 1 + p3 1 + cos 2 e E2 5/2+ 3 E 3 " P7 1 + 6 cos 2 e + cos 4 e M2 3/2" 2 M,- P 5 1 + cos 2 e M2 5/2" 2 M.+ P5 1 + 6 cos 2 0 + 5 cos 4 e E3 5/2- 2 E 2 + p5 5 + 6 cos 2 6 + 5 cos k e M3 5/2+ 3 M 3 ~ P 7 5 + 6 cos 2 6 + 5 cos 4 e P is the center of mass momentum of the pion. - 116 -( Y + p|T|n° + p) - \ T3 - 1/3 T1 + 1/3T0 ( Y + p|T|n + + n) = / | T3 + / | T 1 - /-| T° (37) ( Y + n|T|n- + p) = / 4 T 3 - T 1 - /|-T0 ( Y + n|T|Ti 0 + n) = \ T 3 + \ T1 + 1/3T0 21 where T and T stand for any of the helicity or multipole amplitudes. For 21 21 example, each of the multipole amplitudes E^ +, has components E^ + > and the correct combination of these must be used in equations (36). The 0 1 0 1 3 3 multi- poles E ^ , M lead to a f i n a l state with I = 1/2, and E^ +, to a state with I « 3/2. The four reactions are determined by only three amplitudes T* for each h e l i c i t y state. As the electromagnetic Interaction i s weak compared to the strong nuclear interaction, i t may be assumed that the y-N channel does not contribute to nN elastic scattering, except near the threshold due to mass differences, and further that the transition matrix elements for elastic YN scattering are much smaller than those for photoproduction. With these assumptions the reaction matrix, which is real and symmetric, for the coupled y-U and TC-N channels takes the form: - 117 -Y-N E M 1=1/2 -N 1=3/2 E Y-N M o o o o x l / 2 yl/2 x3/2 y3/2 1=1/2 n-N x l / 2 x3/2 tan b\i2 o 1=3/2 yl/2 y3/2 o tan 6\i2 The submatrlx connecting the two n-N channels Is diagonal in the isospin and diagonal elements are the tangents of the real phase sh i f t s . The matrix elements x* and connect the two isospin states of TI-N system, with electric and magnetic multipole photon states respectively. To f i r s t order in x* and y*, the transition matrix elements are: E A ± ( I ) = x 1 cos 6*± e l ± M A ±(I) = y cos 6 A ± e where E^ +(I) and M^ +(I) are those parts of multiple amplitudes that lead to a f i n a l state of isospin I, i . e . E ^ 1 and M0^1 for 1 = 1/2, and E 3 + , M 3 ± for transitions to I = 3/2. E ^ 1 , and M^'1 have the phase 6*{ 2 so that: - 118 -_0,1 1-0,1. /•.-1/2-1 „0,1 iwO.li r . . l / 2 i . _3/2 M3/2 , „. . .3/2 while , have the phase : E L - 'El± i e x p i*li2) M 3 ± = |M3± | exp ( i 6 A 3 ± / 2 ) If the phase shifts 6* + are known from an analysis of TC-N scattering data, the number of parameters required to describe photoproduction is reduced by one-half. Conversely, the analysis of photoproduction can be used as an Important alternative method of identifying the TC-N resonant states i f the photo-production data are good enough. IV.7.3 The isotensor term of the electromagnetic current Nuclear electromagnetic currents are usually assumed to transform under rotation in isospin space like the sum of a scalar and a vector term: . .s , .v -V -v Ju For total charges, the assumption Is a consequence of the Gell-Mann-Nishijima relation. In nuclei, the existence of an additional component, i.e. the - 119 -i s o t e n s o r term, can be t e s t e d by l o o k i n g f o r the forbidden | A l | = 2 electromagnetic t r a n s i t i o n . The r e s u l t s i n some l i g h t n u c l e i reported by Marrs et a l . (MAR75, MAR76) gave no evidence f o r the i s o t e n s o r term. P o i n t i n g out that the i s o t e n s o r term may be more apparent outside purely nuclear processes, Dombey and Ka b i r (D0M66) and G r i s h i n (GRI67) suggested a study of the photo e x c i t a t i o n of the A(1232) is o b a r as a p o s s i b l e t e s t of the i s o s p i n s t r u c t u r e of electromagnetic c u r r e n t s : yp -* A + n + n yn •*• A 0 -»• it~p Any s p i n amplitude of these r e a c t i o n s may have the f o l l o w i n g s t r u c t u r e A(yp ->• Ti +n) = (3T° + T 1 - T 3 + / j T) A(yn -• Tt -p) - ^ | (3T° - T 1 + T 3 + / | T) where i n a d d i t i o n to the u s u a l components, one a l s o has the Isotensor, AI=2, component T f o r 1 = 3/2 s t a t e . For TC° photoproduction the amplitudes are: 1 T A(yp -• n°p) = j (3T° + T 1 + 2T 3 - / j T) A(yn -»• n°n) = 1 (-3T0 + T 1 + 2T 3 +2 / | T) - 120 -Clearly, to test the validity of the j A l | < 1 rule, one has to perforin pion photoproduction experiments both on proton and neutron targets and compare the results obtained in these processes. In 1971, Sanda and Shaw (SAN71) proposed a test for the |Al| < 1 rule, which is commonly referred to as the 'isotensor dip test'. Assuming that in the (3,3) resonance region the amplitudes may be written as the sum of slowly varying and real non-resonant components and the rapidly varying resonant magnetic dipole and assuming the Watson theorem to be valid, they pointed out that the observation of a dip at the resonance in the difference A'(to) = ^ (cf t o t(Yn •*• Tt-p) - o t o t (YP * +n)) should be the evidence for the existence of the isotensor component T. This dip would be caused by the interference between the resonant M 3 + and T terms, but In the absence of Al = 2 terms, A'(co) should have a smooth behaviour. IV .7 .4 Time reversal Invariance The simplest way to investigate time reversal non-invariance In hadronic electromagnetic interactions is to look for breaking of detailed balance in particular two body reactions. If the yNA vertex were responsible for T- violation, the simplest reactions to be studied would be: yn •*• TC p, yd •* np, y°He •*• dp - 121 -The f i r s t reaction was suggested by Christ and Lee (CHR66). A practical model was developed by Donnachie and Shaw (D0N72) to investigate T-invariance. They incorporated in their model both isotensor and T-violating terms and allowed the latter to be both in the isovector and isotensor resonant amplitudes, so that the magnetic dipole M3 + may be parameterized as follows: pM 3 1 + = - j A(co)xp e p - /-j A(to)(x2e - /y x 3e ) /l i*n "2 i X l» 7 i x 5 nM 3 1 + = -y A(to)xn e = /-j ACioKxje + /y x 3e ) where the subscripts "p" and "n" refer to the charged pion photoproduction on protons and neutrons, X j and x 3 are the isovector and isotensor coupling strength to the resonant amplitude A(co) which is strongly dependent on the cm. energy 10. The T-violating Isovector and isotensor phases x 4 and x 5, as well as <p and (p change sign under time reversal. These changes of sign P n would reveal themselves experimentally in the interference of the resonant amplitude with the standard non-resonant amplitudes. With this model, Donnachie and Shaw found that agreement with experimental data available at that time would imply the presence of both isotensor and T-violating phases: they found x 3 /x2 = -0.28 and (p^  = -11°. However, in order to reach any firm conclusion about a possible violation of T-invariance, one has to have reliable data for both processes yn * Tt -p. Of course the di f f i c u l t y is that - 122 -for yn •*• n~p there w i l l always be the problem that the experiment has to be performed on a deuteron target and i t is impossible to eradicate a l l effect of the spectator proton. This w i l l be discussed further in the next chapte - 123 -CHAPTER V Experimental Results and Discussions V .1 Introduction In this chapter, the f i n a l results for the differential and total cross-sections of radiative capture and charge exchange reactions are presented. The differential cross-sections of the (it- + p •> y + n) reaction are compared with previous data and with several calculations. From the charge exchange diff e r e n t i a l cross-sections, which are obtained by unfolding the energy spectra of the n° y-rays, pion-nucleon phase shifts and scattering lengths are deduced and compared with those obtained from the elastic channels and the phase shift analysis predictions. The charge exchange differential cross-section at zero degree is also calculated. It should be mentioned that when comparing with the previous data, we have not mentioned very poor quality data, but have chosen the most recent, which have better and more reasonable precision. Finally, a discussion of time reversal and isospin symmetries in the pion-nucleon system is presented. V.2.1 Radiative Capture Results The original goal of this project was to investigate the reaction ( u - + p -»• y + n) which contributes significantly to the general study of the photomeson reactions . The highest beam energy at which the experiment could be performed was 125 MeV. Above this energy i t became progressively more d i f f i c u l t to distinguish the radiative capture reaction from the competing - 124 -charge exchange reaction. Angular distributions are available for 30°<8<140° for several energies from 50 MeV to 125 MeV. The differential cross-sections for the reaction (y + n •*• it" + p) are illustrated i n Fig. V.l - Fig. V.6. The numerical values of the (u~ + p ••• y + n ) cross-sections have been transformed to the time-reversed reaction using the principal of detailed balance and are listed in Table V . l . For the moment we assume time reversal invariance and use detailed balance to relate the two cross-secions. The figures also show the predictions of Blomqvist and Laget (BL077, LAG81), Arai (ARA80), and Smith and Zagury (SMI79). Figure V.7 shows the total (n~p -»• yn) cross-sections at different energies and compares them to the calculations, as well as to the most precise measurements in the literature. The previous data come from both radiative capture experiments as well as photoproduction experiments over the (3,3) resonance region. There are many data which have been obtained i n photoproduction experiments, so we have selected the most recent ones to il l u s t r a t e the situation. We w i l l now discuss the existing data which can be divided into two groups: 1) The charged pion photoproduction data; 2) The radiative capture data. V.2.2 The Photoproduction Data In the past fifteen years several improvements have been made in experiments which measured it" photoproduction on neutrons around the f i r s t resonance: 30 20 —i 1 1 1 yn-* w~p TV • 45.6 MeV Ey > 194 MeV T 1 r-FIT B B U I I B a L (2) . ARAI — s az I _l_ 30 Figure V.l - V.6 60 90 9 (cm) 120 150 180 The differential cross-section for the (rc-p -»• yn) reaction shown as the time reversed reaction (yn + TC~p) at six different energies. Also shown are the calculations by (SMI82), (BLA80), and (ARA85). [B & L(l)] is the result of a multipole analysis while [B & L (2)] shows the result of a pseudoscalar-relatlvistic calculation. (S & Z) and (ARAI) are two other multipole analyses. - 126 -Figure T.3 ARAI 11 i i I i i I i i I I I I 1 1 1 1 1 0 30 60 90 120 150 180 9 (cm) Pigar* ?. 4 9 (cm) Figure T. 5 180 - 131 -Table V.la Experimental results for the reaction ( T t - p •*• -yn) [quoted as cm. cross sections for the inverse (yn T f - p ) reaction] Normalization error is 3%. H (Wb/sr) Angle Total Angle Total Lab angle T = 45.6 MeV Tf W = 1116.8 MeV dependent error error T - 62.2 MeV Tt W - 1130.7 MeV dependent error error 30° 8.3 0.3 0.4 7.6 0.3 0.4 45° 8.5 0.3 0.4 9.2 0.4 0.5 60° 9.0 0.4 0.5 10.9 0.5 0.6 75° 10.3 0.4 0.5 11.9 0.5 0.6 90° 11.1 0.5 0.6 12.5 0.5 0.6 105° 13.3 0.6 0.7 120° 14.5 0.5 0.7 15.2 0.7 0.5 135° 16.4 0.6 0.8 141° 13.3 0.6 0.7 16.0 0.6 0.8 a (ub) Total 138.0 5.6 7.0 153.0 5.3 7.0 132 -Table V.lb Experimental results for the reaction (ir~p yn) [quoted as cm. cross sections for the inverse (yn •»• ir~p) reaction] Normalization error i s 3%. of ( u b / s r ) Angle Total % C b/sr) Angle Total Lab angle T - 76.4 MeV it W - 1142.5 MeV dependent error error T - 91.7 MeV W - 1154.9 MeV dependent error error 35° 7.5 0.4 0.5 45° 9.3 0.5 0.6 9.2 0.5 0.6 60° 11.0 0.7 0.8 13.4 0.8 0.9 75° 13.0 0.8 0.9 16.9 0.9 1.0 90° 14.9 0.9 1.0 17.3 0.8 1.0 105° 15.1 1.1 1.2 17.4 1.0 1.1 120° 16.0 1.1 1.2 18.7 1.1 0.9 135° 17.9 1.3 1.4 141° 17.0 1.3 1.4 21.4 1.2 1.4 a (ub) Total 163.0 3.5 6.0 186.0 4.0 7.0 - 133 -Table V.lc Experimental results for the reaction (Tf -p yn) [quoted as cm. cross sections for the inverse (yn -*• Tf_p) reaction] Normalization error is 3%. % ("Wsr) Angle Total Angle Total Lab angle T - 106.8 MeV n W - 1167.1 MeV dependent error error T - 121.9 MeV Tf W - 1179.2 MeV dependent error error 35° 9.0 0.4 0.5 10.3 0.7 0.8 45° 10.6 0.5 0.6 13.8 0.7 0.8 60° 14.9 0.8 0.9 19.0 0.9 1.1 75° 19.6 1.0 1.2 20.7 1.0 1.2 90° 20.1 1.0 1.2 24.3 1.5 1.7 105° 22.2 1.1 1.3 23.8 1.3 1.5 120° 22.4 1.3 1.5 27.1 1.7 1.9 135° 17.9 1.3 1.4 27.9 1.6 1.8 141° 22.8 1.4 1.6 23.5 1.7 1.9 o (ub) Total 216.0 4.7 8.0 250.0 6.6 10.0 \ . ( M e V ) 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 T 1 1 1 1 r~ 1 0 5 0 1 1 0 0 1 1 5 0 1 2 0 0 1 2 5 0 1 3 0 0 LJ ( M e V ) Figure V.7 The total cross-section of the (yn + it~p) reaction as a function of energy including previous experimental results and the theoretical results of Blomqvist and Laget (BL077). - 135 -a) First let us consider the measurements of the production ratio of i t ~ / n + on deuterium where the deuteron structure effects are expected to cancel out. Such measurements were performed at the Universities of Bonn (VON74) and Tokyo (FUJ77), both using magnetic spectrometers to detect the T I + and iC under the same kinematical conditions. The (yn -»• n~p) cross sections were obtained from the reactions: y + d •»• n+ + n + n (1) y + d ->• i t ~ + p + p (2) by multiplying the ratio R(^+), corrected for the f i n a l state Coulomb inter-action and Pauli exclusion principle, by the measured (yp •*• n +n) data (FIS72, FUJ77). The results of Adamovitch et a l . (ADA69) are an analysis of the world data existing at the time using the ratio method. b) At higher energies, there are data obtained by a German collaboration (BUT70) in the DESY bubble chamber, where the impulse and closure approxima-tions were used to extract the (yn •*• n~p) cross-sections from the reaction yd -»• it~pP (BEN73) . c) The f i r s t i t " production cross-section using monochromatic tagged photons became available with the work of Boucrot et a l . at Orsay (BOU73). In this experiment the deuteron target was placed inside a magnetic spectrometer equipped with MWPC, detecting both the outgoing n~ and the recoiling proton. Kinematical reconstruction selected events close to the free neutron domain. - 136 -d) A detailed and thorough analysis was made of the data from a high sta t i s t i c s counter experiment performed at Saclay (ARG78) in which the yd •+ Tt-pp reaction yield was measured under well-defined kinematical conditions. The high intensity, photon beam of the Saclay linac was used, and the pion and one of the outgoing protons were detected in coincidence in a co-planar geometry, and their momenta p^ and p were measured. Therefore, the remaining kinematical variables, the photon energy Ey, the momentum p f and the polar angle Q f of the undetected nucleon were determined. This complete knowledge of the kinematics allows one to check the validity of the spectator nucleon model and to look for deviations. The invariance of the extracted elementary cross-section (yn •*• n _p) with respect to p f and/or 9 f is a crucial test of this model. The published results for (yn -*• n -p) (excitation curve at 9 = c .m. 90°) were extracted in a domain with a low value of the detected nucleon momentum where the spectator models were found to be valid. It has been found that when the recoiling nucleon momentum increases (p f = 150 MeV/c), significant departures from the spectator nucleon model are found which are presumably the signature of the f i n a l state effects. The important conclusion of this work is that the events with low momentum values of the recoiling nucleon (p y - 50 MeV/c) can be selected to extract (yn •* n""p) results. F i g . V .8 compares the extracted cross-sections at 90° to the data obtained from the bubble chamber experiments and from the measurements of the - j ^ . r a t i o . F i g . V.9 also compares the same result with the data deduced from the inverse reaction (n~p •-»• yn) (GUE75, BER74). In another experiment (FAU84) this group has investigated pion production on hydrogen and deuterium in the A resonance - 137 -y D - » p p n " 30L a 2 3 o 5 10. y«-rwp*TT" CJ-90" • NFPR oABBHHM • BONN i "~' 1 i . 1100 1200 1300 Q ( T i N ) ( M e V ) Figure V.8 The comparison between the (yn Tt""p) reaction cross-section (shaded area) extracted from (yd •*• ppit -) yield when P r = 50 MeV/c (ARG78), and the bubble chamber data ( f u l l circles)(R0S73), (open circles) (BEN73), and those deduced from the (Tt +/Tt -) ratio measurements (open triangles) (H0L74, FUJ77, FUJ72, CLI74). - 138 -y-f rwp+TT" o=90* 1100 1200 1300 Q(nN)(MeV) Figure V.9 The comparison between Saclay data, shaded area (ARG78), and the cross-section deduced by detailed balance, from the (u~p -*• yn) cross-sections ( f u l l c i r c l e s ) (GUE75), (open triangles) (BER74). - 139 -region using quasi-monochromatic photons. Their experimental data agree within 15% with the predictions of a model which include treatment of the N-N f i n a l state interaction everywhere except in the region near 300 MeV where 20% discrepancies are observed at forward angles. It is claimed that the discrepancies stem from an overestimation of the N-N rescattering contribution by the model. Comparison of the different results suggests that there are s t i l l discrepancies among the i t - photoproduction data. As far as the bubble chamber data are concerned, the reported Frascati results (ROS73) are smaller at backward angles compared to the DESY results (BEN73). This difference arises from the different methods used for extracting the (yn •»• Tt-p) cross-sections. For example, the Frascati results are obtained using the spectator model without any further correction. Although the counter experiments of Bonn and Tokyo agree with each other and with the DESY results, they d i f f e r strongly from the data of the Orsay group: at 327 MeV, the Orsay angular distribution exhibits a pronounced dip at 180° in agreement with the Frascati group results. At this point some discussions of the deuterium data and the way that the photoproduction results are extracted become necessary. As i t was mentioned before, the simplest thing to do Is to observe reactions (1) and (2) and then take the experimental -p. ratio, correct for Coulomb effects, and then obtain the neutron cross section from the formula f j . C T. * O ) - K C £ > £ (YP * *+ • .) - 140 -Now In the application of this equation there is a d i f f i c u l t y ; since we are dealing with a three-body f i n a l state and observing only one body, the pion, we observe a continuous spectrum for monoenergetic incident photons. This i s illustrated in Fig. V.10, which indicates the outgoing n + and it" energy spectra. In these figures, the peak at the highest pion energies corresponds approximately to two body kinematics with the nucleons going off in a correlated way, since the di-nucleon state is not bound. One notes that the shapes of the n~and n + spectra are not identical, which reflects a Coulomb difference, and possibly more, in the f i n a l state interaction. Another example of the three body kinematics in the f i n a l state is shown in Fig . V . l l . In this case the pion energy and angle were kept fixed and the photon energy was varied. Again the same kind of physical effects can be seen. With no f i n a l state interaction (FSI) the pion spectrum is predicted to be featureless, a kind of bell-shaped curve. Taking the attractive N-N FSI into account gives the peak at low photon energies, which is analogous to the peak at high pion energies in F i g . V.10, where the N-N pair tend to get out together at low relative energy. The main difference between the curves shown in Figs. V.10 and V . l l li e s in the way that the kinematics were arranged, and also, that one is in the A energy region and one is at a low energy (T^31 29 MeV). In taking the i t ~ / u + ratio i t is not clear how much of the pion energy spectrum one should include, and in fact in many experimental articles on the subject this information Is not even presented. Another problem is that for bremsstrahlung photon beams there is an additional uncertainty about the i n i t i a l photon energy, which leads to a smearing out of the pion energy - 141 -Figure V.10 Preliminary results for the spectra of the pions emitted in the reactions (yd •* nnn +) and (yd •*• ppn~) at (8 7 t)i ab ™ 45.6°, when Ey » 299 MeV. The broken line curves include only the quasi free contribution, whereas the f u l l line curves include also the f i n a l state Interaction. The (yp •* it +n) reaction indicates the momentum resolution (LAG81). - 142 -Figure V.ll Differential cross-sections for D(y ,ir ) reactions versus photon energy for 29 MeV pions at © 1 ^ - 90° (FRA80, BER81). The cross hatched represents the experimental results. The line marked no FSI is the calculation without final state interaction. The curve narked B - • is for a zero range N-N interaction. The solid curve is for a finite range N-N Interaction. - 143 -spectrum. This latter d i f f i c u l t y has been avoided by the use of tagged or "monochromatic" photon beams at Tokyo and Saclay. Data for the i t ~ / n + ratio from these groups are presented In Fig. V.12, together with calculations of Laget (LAG81). From the calculated curves i t can be seen that the free n ~ / i t + ratio calculated with the elementary theoretical cross sections of Blomqvist and Laget (BL077) is considerably modified by the deuteron structure effects, and then again by f i n a l state interaction effects. These include the antisymmetrization of the identical N-N f i n a l state as well as the f i n a l state N-N Interaction. To be more specific, one must ask what new features enter here that do not occur for the nucleon target. The f i r s t is kinematical. The target nucleons are in an eigenstate of energy but not momentum. Therefore, the production amplitudes must be evaluated for kinematical situations which are not allowed for free nucleons. In addition, there are important dynamical corrections to the impulse approximation (by the impulse approximation, one means the neglect of any dynamical modifications of the elementary operator by the nuclear medium), which can be thought of in terms of rescattering of the pion and the two nucleons In the f i n a l state. From Fig. V.12 one can see that the ratio differs from the impulse approximation by =20%. Therefore, one can assume that to this order of accuracy the (yn -* n~p) cross sections obtained from the ratio procedure w i l l be adequate, but for a greater accuracy some refinements w i l l have to be made. Confirmation of this can be obtained in several different ways. First we can - 144 -T — i — i — i — i — I I — i — i — i — i — i — | I — i — i — i — i — r — i — i — i — i — i — I I i i i i i I I i i i i i 30 90 150 30 90 150 30 90 150 G ) ( c l e g ) Figure V.12 The T t _ / n + ratio angular distributions when Ey = 260 MeV, Ey - 300 MeV and Ey • 350 MeV. The experimental results are from Saclay (FAU84, f u l l circles) and Tokyo (FUJ77, open c i r c l e s ) . The broken line curve is the free nucleon T E _ / T C + r a t i o . The dash-dotted line curve includes only the quasi-free amplitude. The f u l l line curves also take into account the rescattering diagrams (LAG81). - 145 -compare the elementary cross sections obtained from the deuterium targets with those obtained from the inverse reaction (n -p -*• yn). The results have been shown in Fig. V.7. In general, there is as good agreement between our results and the inverse reaction results as they agree among themselves, thus detailed balance does not appear to be violated. However, i t should be remembered that different methods for extracting the (yn •* n -p) cross section give results which vary by 20% or more (BER81). Thus an improvement In the experiment entails much more detailed investigations of the parameters of each event and therefore considerably higher stati s t i c s would be required, a problem which has been discussed in detail by Ramachandran et a l . (RAM79). V.2.3 The Radiative Capture Data The data for radiative capture were measured in the laboratories of Saclay (MIL77), CERN (TRA79) and LBL (C0M75). A l l of these experiments detected both the neutron and the y-ray to separate this reaction from the competing charge exchange reaction. Because one is detecting two neutral particles with uncertain efficiency, the errors are typically larger than 10%. The preliminary results of the UCLA group (BER71) which measured the angular distribution for equivalent photon energies of 335, 481, and 519 MeV, provided at that time, the main impetus for a renewed interest in time reversal violation in the electromagnetic interactions. The results were revised in 1974 (BER74), and, more recently, extended to cover the whole energy region of the P33 resonance (C0M75) . It should be mentioned that for Ey = 360 MeV, the results of the two UCLA experiments do not fu l l y agree among themselves, - 146 -particularly at backward angles. The CERN experiment measured the radiative capture of n~ on hydrogen from P^ _ - 210 MeV/c to P^. - 385 MeV/c and for cm. angles from 30° to 120°. The CERN data are, in general, In fa i r agreement with those of Comiso et a l . , except for G C" m - 60° at E - 285 MeV and 360 Y Y MeV. They have also good agreement with the energy independent multipole analyses (PFE72, SUZ74, BER75, GAN76). Unfortunately, in the low energy region (T < 140 MeV), i t is very hard to find data of good precision, because of the problems which were mentioned in Chapter One. Therefore, i t is d i f f i c u l t to compare our results with previous radiative capture data. However, we have good agreement with the better quality data in general and with the CERN data in particular. At lower energies our results are somewhat higher than the Saclay results of Balestri et a l . (BAL77). However, i t should be noted that the f i n a l version of their analysis has never been published. V .3 Comparison with Multipole Analyses and Other Theoretical Calculations The various published multipole analyses of pion photoproduction (BER68, ENG68, SCH69, BER69, WAL69, N0E71, PFE72, BAL73, SUZ74, M0R74, BAR76, BAR78, SMI79, ARA80, CRA80) contain some adjustable parameters which are determined from a f i t to the photoproduction data existing at the time, and include (yn + n~p) data obtained from measurements with a deuterium target. Therefore, the agreement and disagreement of our data with multipole analyses reflects the consistency of our data with the data used in the specific analysis. Phenomenologically, the scattering amplitude for photoproduction can be written as a sum of contributions from a l l possible multipole transitions - 1 4 7 -characterized by multipole order j , total angular momentum J and parity p. Let a summarize these indices. Each multipole has a complex amplitude given 16 by M e , where M and 6 are real numbers. The differential cross section oc a a may be written: I M f (x) + T M M cos (6 - 6 D)f D(x) dQ ^ a a % a B v a B a B a<6 r and the polarization $ of the recoil nucleon as: (X f p CQ X to where x - cos 0 , 9 = the pion angle in the center of mass, and n = — | q x £ | where q and £ are respectively the momenta of the incoming photon and the outgoing pion. The functions f (x), f Q ( x ) , and g „(x) are polynomials a otp ap determined by angular momentum and parity associated with the multipoles a and In the analysis of the region W < 1300 MeV only s and p waves are considered, and only a limited number of multipole amplitudes are actually determined from experiment, namely the electric dipole E Q + , magnetic dipoles M1+, and E 1 + amplitudes. The higher partial waves are evaluated from theory. It is assumed In a l l analyses that the phases of the multipoles are given via the Watson theorem from their it-N scattering phase-shift values. Multipole analyses of single pion photoproduction are discussed by Tran et a l . (TRA79). Among them we have chosen the most recent ones (ARA80, ARA82, CRA80, CRA83, SMI79) for comparison with our data. - 148 .-a) The Tokyo Analysis This is an energy dependent partial wave analysis of single pion photo-production from the f i r s t resonance region through the fourth resonance region in which 7768 data points on the processes (yp -»• it +n), (yp + it°p), and (yn •*• 7t _p) have been used. The method employed in this analysis is similar to that of Moorhouse et a l . (MOR74) which is specified by using fixed-t dispersion relations and the K-matrix formalism, where the K-matrix is parameterized in the form of resonance poles plus background. In addition to this framework, a Regge-like parameterization (BAR74) of the amplitudes was employed in the higher energy region and the partial waves from s-wave up to h-wave were included. The data used in this work are mainly based on the data compilation of Nakamura and Ukai (NAK77). One of the attractive features of Arai's analysis is that i t contains substantially new data points for the process (yn •* n~p). As for the differential cross section, this analysis systematically reproduces the experimental data rather well. However, there are several discrepancies at small angles for the processes (yp -• it +n) and (yn •»• u~p). As for the polarization and asymmetries, there is very good agreement between the predictions and the experimental data. As can be seen in Figs. V.l - V.7, we have also excellent agreement with the most recent version of the Tokyo analysis at a l l energies and angles. However, a Legendre polynomial f i t to our data does not agree with Arai's predictions below 30 degrees, especially at lower energies, and no data exist to define the amplitudes - 149 -In this region. The present experiment was not able to measure these angles because of the muon halo around the pion beam. b) Multipole Analysis of Smith and Zagury This is an energy-dependent multipole analysis of photoproduction of pion from neutrons and protons from threshold up to 450 MeV (SMI79). The experimental results were taken from the data compilation of Menze, P f e i l , and Wilcke (MEN77) and the TC-N phase shifts from Almehed and Lovelace (ALM72). The data used for photoproduction from neutrons are quite poor and much fewer than those for a proton target. For example, in the Menze compilation, in the region of photon laboratory energies from threshold up to 450 MeV there i s no report of nucleon-recoil polarization measurements, and just one result for a polarized target experiment. In this work i t is assumed that j > 3/2 multipoles are given by the Born terms and other multipoles are determined using, as a f i r s t approximation, a dispersion-relation model and corrected by the product of three terms, i.e.: a) a phase factor as given by the Fermi-Watson theorem; b) threshold behaviour dependence; and c) a second degree polynomial in energy (three adjustable parameters for each multipole) . As far as our results are concerned, there is some disagreement with this analysis, which Is probably due to the fact that the selected yn •*• TC-P data are of poor quality, having too low a cross section. - 150 -c) The Glasgow yNN and vNA Partial Wave Analysis This is basically a partial wave and amplitude analysis of single pion photoproduction for reactions (yp n +n, yp • "^p, and yn + n -p) at a l l energies between threshold and a photon laboratory energy of 1.6 GeV (CRA80, CRA83). It is similar to the 1978 Glasgow analysis (BAR78) and i t s method consists of a multipole analysis at low energies, an amplitude analysis at high energies and a combination of both at intermediate energies (2 < W < 2.5 GeV). In general the f i t is good for an energy dependent analysis with the poorest results for the yp Tt+n reaction. V.3.2 Comparison with Other Theoretical Calculations Aside from multipole analyses, we have compared our results with calculations of Woloshyn (W0L84) and predictions of Blomqvist and Laget (BL077, LAG80). The present data do not agree with Woloshyn's calculation which is based on the chiral Invariant effective Lagrangian model of Peccei (PEC69). Woloshyn's results are not shown in the figures, but they are close to and follow the pattern of Blomqvist and Laget (2). The results of Blomqvist and Laget are shown in Figures V.l - V.7, together with the rest of the experimental results which have been discussed. It is clear that our data do not agree with their calculations which are based on the f i r s t two Born terms and a A formation amplitude. The calculations offer three variants of the model they use, i.e.: (a) the pseudo-scalar Born terms plus iosbars, non-relativistic calculation; (b) the pseudo-vector Born terms plus isobars, r e l a t i v i s t i c calculation; and (c) the pseudo-scalar Born terms plus isobars, r e l a t i v i s t i c calculation. Fig V.13 shows the different Feynman diagrams - 151 -ib BORN PS A BORN PV - x - x 1d "5. Figure ¥.13 The different diagrams considered in pion photoproduction on nucleons. The wavy line, broken line, and full line represent respectively the photon, pion and nucleon. The A(1236) Inter-mediate state is represented by a double line, (a) and (b) are the PS Born terms and PV Born terms plus S-channel A(1236) formation, (c) and (d) are time ordered decomposition of direct and crossed nucleon Born terms (BL077). - 152 -considered in pion photoproduction on nucleons. Even though the agreement between theory and experiment is quite good above the resonance (Fig. V.7), a l l of the variants l i e above the global data In general and above our data i n particular, below resonance. At this point i t should be mentioned that the data from the f i r s t phase of this project (SAL84), i . e . at T^= 27.4 MeV and 39.3 MeV, are in good agreement with the calculations of Blomqvist and Laget but somewhat higher than the Saclay results of Balestri et a l . (BAL77). This agreement at very low energies can be explained in terms of the use of the low energy theorems. That i s , these calculations are limited by the constraints of low energy theorems and should agree with low energy data. So, i t is clear that although the general trend is correct, there is a clear difference between data and calculations for the relative weighting of the El term for the i n i t i a l shoulder and the Ml term for the A resonance, and i t seems that the Ml contribution is overestimated. These amplitudes are the standard for calculating photomeson production on nuclei, and since these calculations do not agree very well with the experimental results, i t seems that the elementary interaction below resonance is not as well understood as i s commonly assumed. Aside from the conventional calculations, our data offer an opportunity for a new line of calculations and a test for different quark models. For the (n -p + yn) processes, the yn vertex is a three quark-gamma vertex and there is no ambiguity In doing the calculations, as i t has been done before. The 7t~p vertex is a four quark-antiquark vertex and therefore the calculations are model-dependent. Hence, proceeding via different quark models, for example, the bag model, etc., and comparing the f i n a l results with - 153 -experimental data, w i l l offer a test of the assumptions employed in the various calculations (ARA84, KAL83, KAL84). V .4 Time Reversal Invariance Experimental tests of time reversal invariance (TRI) have acquired considerable importance because of the observed breakdown of CP invariance i n K° decay in the weak interactions . CP violation was discovered by Cronin and Fitch and is now well established (BER85, BLA85), thus time reversal is also violated. The reason for this is that the T and CP transformations are connected by the famous CPT Theorem. The proof of this theorem is based on very general assumptions and is cherished by theorists in the sense that i t is d i f f i c u l t to formulate f i e l d theories which are not automatically CPT invariant, because basic beliefs like causality cannot be included. Until 1964 i t was believed that a l l types of Interaction were invariant under the combined CP operation. The weak interactions were known to violate C and P Invariance separately, but thought to respect the combined CP symmetry. However i t was discovered (CHR64) that the neutral long-lived K particle, which normally decays by the weak interactions into three pions of CP eigenvalue -1, could occasionally decay into two pions, of CP = +1. The origin of CP violation i s . s t i l l not understood, but whatever the process i s , i t implies a T violation because of the CPT theorem. It is likely that the effect arises from a specific type of weak or superweak interaction, but because of the uncertainty in i t s origin, searches have been made for evidence of T-violation in the weak, electromagnetic and strong interactions. Time - 154 -reversal violation would imply a transverse polarization of the muon in the weak decay (IT*" -»• n° + u + + v) relative to the T C ° , V plane. Several recent experiments have failed to detect any transverse polarization. This would have been revealed as an up-down asymmetry of the decay electron relative to the above plane, In the subsequent decay (p,+ •*• e + + v + v ). e [i The simplest way to investigate time reversal non-invariance in hadronic electromagnetic interactions is to look for a breaking of detailed balance, in two body reactions. As far as the yNA vertex is responsible for T violation, the simplest reactions to be studied would be (yn *• Tt"p), (yd -*• np), and (y3He + dp). A recent confirmation of T-invariance at the yNA vertex came with the investigation of Heusch et a l . (HEU76), who studied the two body photo disintegration of the 3He and i t s inverse processes In an energy region which covers the excitation of the A(1232) isobar from one nucleon. However, not a l l electromagnetic interactions are suitable for testing TRI since current conservation alone w i l l assure detailed balance when the nucleon involved Is on the mass shell (BER65). This restriction is removed when the nucleon is excited to a resonance. As the A(1232), and possibly other nucleon resonances, play an important role in photonuclear reactions on light nuclei above E^ , = 200 MeV, this may well be an ideal place to test TRI (CAM84). A review of photodisintegration and radiative capture on light nuclei (A = 2, 3, and 4) is given by Cameron (CAM84). Further confirmations come from the work of Tran et a l . , who studied the radiative capture of pions on hydrogen in the A(1232) region, and the present limits on the neutron electric dipole moment - 155 -(RAM81). A direct comparison of the (it~p -*• yn) measurements can be made with the (yn •* it~p) cross section determination of Adamovitch et a l . (ADA69). The two sets of measurements are in close agreement, and an upper limit of about 4% can be set for any time reversal non-conserving components of this interaction. A more sensitive test of time reversal non-conservation is to be found in a comparison of the shape of the differential cross sections for the two reactions, due to the interference of different reaction amplitudes (DON72). - 156 -V.5.1 The Charge Exchange Results For this reaction, angular distributions are available for 30° < 0 < 140° for several pion kinetic energies from 50 MeV to 125 MeV. Above T^ - 125 MeV, i t becomes progressively more d i f f i c u l t to distinguish the radiative capture reaction from the competing charge exchange, so i t was decided to halt data taking. The y-ray spectrum at any angle actually contains information on the f u l l angular distribution of the i t 0 , therefore there is much redundancy among different angles for the y-ray detector. In fact, the extreme forward or backward angles are the best, but unfortunately for angles less than 30°, there is a serious background which originates from the decay muons (n~ -*• H~v^) in the beam, and for angles greater that 150°, i t was mechanically impossible to position the crystal without interfering with the last quadrupole element in the beam l i n e . An additional complication also occurred for the lower energy pion beams, vi z . a few of the pions scattered in the target and then stopped. Unfortunately, a stopped i t - always gives off a y-ray whereas only 10~3 of the passing pions produce a y-ray. Thus, i f only 10~3 of the incoming flux stop in the target, there w i l l be a contamination in the y-ray spectrum equal to the in-flight events. F i g . V.14, in which the data were obtained using the low energy pion channel M13, shows clear evidence of stopped i t " interactions. Therefore, i t is necessary to not only do a l l of the processes previously mentioned before f i t t i n g the in - f l i g h t n° spectra, but also to optimize the f i t for the stopped i t " spectrum. As the characteristic spectrum for a stopped E y ( M e V ) Figure V.14 Y-ray energy spectra at 27.4 MeV and 90°. One can Identify clearly the (ii~p •*• yn) and (*~p > n°n) events both for stopped and In-flight pions. - 158 -TC~ Is well established, i t turns out that this added complication does not seriously affect the f i t t i n g for the in-flight n~ reaction at low energies, and s t i l l i t is possible to obtain an excellent f i t with a small error. For the higher energies, after f i t t i n g many spectra, i t was also established that there was a few percent contribution from stopped pions. Aside from this, the overall f i t to the energy spectra was broken down into the constituent parts from three different Legendre polynomials in the angular distribution plus an exponential background, and their coefficients were found. The differential cross sections were then obtained by taking a weighted average of the Legendre polynomial coefficients A^ of the TC° distribution obtained at each detection angle. The fin a l results are presented in Table V.2 and are plotted in Fig. V.15 and V.16. Variations of parameters A^ and A 2 with energy are plotted in Figs. 8^  and B 2 in Appendix B. The total cross section can be obtained absolutely and does not suffer from many of the problems of elastic scattering measurements; for example the y-ray does not decay after leaving the target, etc. The complicated r e l a t i v i s t i c transformations which are necessary in order to go from a it 0 angular distribution to a y-ray energy distribution are quite straightforward and have been reproduced by several groups (BOD57, KER60) and were also employed in the analysis of a similar experiment at CERN (BAY76). The values for the charge exchange scattering length (aj-a 3) are given in Table V.4, together with the previous results. The s-wave scattering length i s determined by f i t t i n g hadronic pion-nucleon amplitudes to the experimentally determined Legendre polynomial coefficients A^. In the energy region under study, only s- and p-waves contribute to the charge exchange cross section, and therefore only these amplitudes were used in the f i t . The TABLE V.2 Experimental Results for the (u~p -*• n°n) Reaction Tn= 45.6 MeV T n = 62.2 MeV Tn = 76.4 Mev T n - 91.7 MeV T n - 106.8 MeV \ - 121.9 MeV AQ(mb/sr) 0.516 ± 0.010 0.654 ± 0.020 0.879 + 0.020 1.26 ± 0.03 1.68 ± 0.08 2.28 ± 0.04 Aj^  (mb/sr) -0.709 ± .020 -0.90 ± 0.03 -1.16 ± 0.03 -1.37 ± 0.10 -1.49 ± 0.10 -1.72 ± 0.15 A2(mb/sr) 0.20 ± 0.04 0.36 ± 0.04 0.50 + 0.06 0.83 ± 0.10 1.27 ± 0.10 1.71 ± 0.15 aTOTAL ( r a b ) 6.48 ± 0.24 8.2 ± 0.4 11.1 ± 0.4 15.8 + 0.6 21.1 ± 1.2 28.7 ± 1.0 The differential cross section is expressed in terms of Legendre polynomials with coefficients A The error quoted for these coefficients does not include an overall normalizing error of 3.1%. However, this error is included for the total cross-section. - 160 -30 60 90 6 (cm.) 120 150 180 Figure V.15 The differential cross-sections for the charge exchange reaction at different energies, as obtained from our best fits. - 161 -1 0 0 1—I I I I 1 — I — I I I I. C o QJ I/) I l/) I/) o OJ c n C ro x OJ OJ c n c— ro Zidell et al E 85 Arndt Karlsruhe Rowe et al. 1 0 J i i i i i I ' 1 0 o X • • • V • Bugg et al. Miyake et al. Spry Roberts and Tinlot Cundy et al. Donald et al. York et al. Carroll Frank Jenefsky et al. Comiso and Berardo Present results i I i i i i i 1 0 0 Pion energy (MeV) 4 0 0 Figure V.16 The total cross-sections for the charge exchange reaction including previous experimental results and recent phase-shift analyses. - 162 -d-wave amplitudes are much smaller and can be neglected. The charge exchange cross section in the center of mass frame can be written as: ^ c x - I V A (CO.0) -|O.[|£3 8 1 / 2 - f l s ^ + < 2 f % 3/2 + f 3 p 1/2 " 2 f l p 3/2 " f l p l/2> C ° S 9 I ' + K f 3 p 3/2 " f 3 p 1/2 " f l p 3/2 + f l p l/2> S i n 9 l 2 ^ ( 1> where p is the momentum of the TC° , p is the momentum of the n -, and the f 0 -parameters are the scattering amplitudes which are related to the phase shifts 21 21 6 by the expression f^ = {exp(2i6^j) - l } / 2 i . Because there are three experimental parameters, i . e . A.^ s, and six scattering amplitudes, standard phase shift values (ARN85, KOC80) are used as the starting point for the f i t . The phase shifts obtained are not the pure nuclear phase shifts 6„ but the effective phase shifts 6 N„ = 6 N + 6 C, where 6^ is the correction due to the distortion of the Incoming pion by the Coulomb potential. In this analysis, we used the known values of the p-wave shifts to calculate the s-wave phase shifts for the six energies investigated. The phase shifts then were corrected for Coulomb effects, following the standard method of Tromberg et a l . (TR077), in order to obtain pure nuclear s-wave phase shifts which in turn determines the difference between the s-wave scattering lengths (a^-a^). - 163 -In general for pure nuclear phase shifts in each eigen channel we can write an analytic function which incorporates the threshold behaviour expected for a f i n i t e range interaction plus a term which represents the nearest nN resonance: tan 6 £ ^2x+l = D + c c l 2 + "resonant term" where q is the TtN center-of-mass momentum. Therefore the s-wave scattering lengths for the charge exchange reaction can be calculated from the formula: tan 6j - tan 6 3 a2 - a 3 = + ( c 3 - c± )q 2 + "resonant term' where q = /q q . The phase shifts are tabulated in Table V .3 and the resulting s-wave scattering lengths are shown in Table V.4, and compared to other values available In the literature (BUG73, DUC73, SPU77). V.5.2. Comparison with Phase Shift Analyses and Other Data Over the last fifteen years a number of experiments have been performed which have considerably improved the precision of the pion nucleon scattering data. These new data have had the effect of making some of the older widely quoted partial wave analyses obsolete and instigated the new ones. Table V .5 summarizes the pion-nucleon scattering data, from 0 to 350 MeV since 1971, which are the foundations of the new partial wave analyses. The most widely TABLE V.3 Phase Shifts (in degrees) obtained from the data after applying the electromagnetic corrections T n - 45 .6 MeV T^ - 62.2 MeV \ ' 76.4 MeV 1„ -TC 91 .7 MeV T n = 106.8 MeV T n - 121.9 MeV 6.31 + 0.38 6.78 ± 0.59 7.59 0.59 8.06 + 0.62 8.03 ± 0.72 8.64 + 0.84 S31 -5.04 + 0.39 -5.87 ± 0.61 -7.49 + 0.63 -8.74 + 0.66 -9.52 + 0.75 -10.79 + 0.91 P l l -1.11 + 0.28 -1.21 ± 0.40 -1.31 + 0.79 -0.97 + 0.48 -1.03 ± 0.52 0.55 + 0.62 P13 -0.41 + 0.20 -0.610 ± 0.34 -0.79 + 0.43 -0.89 + 0.45 -1.18 + 0.61 -1.38 + 0.75 P31 -0.67 + 0.52 -1.070 ± 0.81 -1.595 + 0.93 -1.89 + 0.97 -2.34 ± 1.19 -2.82 + 1.38 P3 3 5.08 + 0.61 8.74 ± 0.70 12.49 + 0.78 18.52 + 2.4 25.22 ± 2.8 34.92 ± 5.26 - 165 -TABLE V .4 The S-wave scattering length (a± - a 3) obtained from this experiment and other references (in natural units TT/m c < al - a 3) \ - 45.6 MeV 0.267 + 0.013 62.2 MeV 0.253 + 0.017 This - 76.4 MeV .0.270 + 0.015 experiment 91.7 MeV 0.273 + 0.015 - 106.8 MeV 0.263 + 0.015 \ - 121.9 MeV 0.270 + 0.017 (SAL83) 27.4 MeV 0.260 + 0.012 (SAL83) = 39.4 MeV 0.265 + 0.012 (DUC73) 0.270 + 0.014 (SPU77) 0.263 + 0.005 (ZID80) 0.302 + 0.006 (ROW78) 0.283 + 0.008 (KOC80) 0.274 + 0.005 (KOC86) 0.275 + (BUG73) 0.262 + 0.004 (ARN80) 0.258 + 0.006 - 166 -TABLE V.5 List of the Pion-Nucleon Scattering Experiments Used i n Phase-Shift Analyses W M e V > Laboratory S t a t i s t i c a l Error (%) Observable Reference 70-295 CERN 1 4 BUG71 70-295 CERN 0.5 + Oj CAR71 298-336 LBL 0.4 + Oj DAV72 142-272 CERN 6 o°(0°) BAY76 207-370 LBL 4 a 0 (6) BER72 21-96 SACLAY 2.10 o + ( 9 ) BER76 40-50 LAMPF 5 o + ( 9 ) BLE78 88-292 CERN 2 c r ( 9 ) BUS 7 3 137-260 LBL 7-9 a 0 ( 9 ) C0M75 23-43 SACLAY 10 a 0(180°) DUC73 27-39 TRIUMF 4 a 0 ( 9 ) SAL 8 4 277-334 LNPI 2.5 c r ( 9 ) G0R76 147-347 PPA 3-6 o ° ( 9 ) HAU71 114-227 CERN 4-14 a°(160 o) JEN74 292-308 SIN . 7-16 P " ( 9 ) ALD78 236 SIN 10-30 P + ( 9 ) AMS75 95-194 SIN 5-30 P + ( 9 ) AMS76 291-310 SIN 10-20 P +(6) DUB77 243-349 LBL 10-15 1^(8) GOR73 70-370 SIN 1-2 e T PED78 65-140 LAMPF 4-6 a + ( 9 ) RIT83 - 167 -used phase shift analyses today are the "Karlsruhe-Helsinki" (KH) analysis (KOC80, HOH79); the Virginia Polytechnic Institute (VPI) analyses (ZID80, ARN85), and the Carnegie-Mellon, Berkeley (CMU-LBL) analysis (CUT80, CUT79). The KH analysis is an energy independent partial wave analysis of pion-nucleon elastic and charge exchange differential cross sections and elastic polarizations . The basic idea of the KH analysis is to exploit as much as possible a l l general theoretical constraints like Lorentz invariance, unitarity, crossing symmetry, analyticity and isospin invariance, and performing tests of the last two properties where i t i s possible. These theoretical constraints are used to find a unique solution for the scattering amplitudes. The question whether the applied constraints are strong enough for this purpose, and which additional assumptions on the asymptotic behaviour for fixed-t amplitudes have to be made, has been investigated by Stefanescu (STE80). The VPI analysis is a comprehensive partial wave analysis of pion-nucleon elastic scattering and charge exchange data below 1100 MeV laboratory kinetic energy. In this analysis, the partial wave structure i s characterized by the location in the complex energy plane of the dominant poles and zeros, which are related to pion-nucleon resonances. Scattering lengths are extracted from the energy dependent solution to characterize the low energy behaviour. In general there is good agreement between the VPI and KH analyses below 600 MeV, and f a i r agreement above 600 MeV. The principal difference between the VPI analysis and that of the KH group is in the larger amount of the dispersion-theoretical results used by the KH group to constrain their solutions. The VPI group uses only the real part of the forward - 168 -amplitude to complement real scattering data. The KH analysis is also much more ambitious, in the sense that i t covers an energy range nearly three times larger than that of the VPI analysis. The main emphasis of the VPI analysis is the encoding of scattering data in i t s more limited energy range through solutions which have proper direct channel analytic properties, and which can be analytically extended into the complex energy plane to reveal dominant dynamical features such as poles and zeros of the resultant partial waves. It is impressive to see that these two quite different approaches produce such similar results. Our total cross section for charge exchange is illustrated in Fig. V.16 in which we compare our results with previous measurements, as well as a selection of phase shift analyses. A comparison reveals that below 70 MeV our results are lower than the average of earlier measurements. However, there is a reasonable explanation for the higher values of the previously measured cross sections. As mentioned before, the rc°-decay spectrum has an important contamination from stopped TC - events, and there is a large magnification of this effect at lower energies. Also, a l l of the previous measurements had extremely poor energy resolution and consequently were not even aware of the effect of stopped pions. As for the higher energies, we have excellent agreement with the accurate measurements of Bugg et a l . (BUG71). Fig. V.16 also indicates that our results are consistent with several phase shift analyses which did not have our results in their data base. In particular, we have excellent agreement with the KH and the latest VPI results. As can be clearly seen in this figure, the analysis of Z i d e l l et a l . (ZID80) gives too high a cross section - 1 6 9 -and scattering length (0*302), even in comparison with older data, whereas the KH value (0.27A) i s in much better agreement with the latest data. Z i d e l l analysed the (n~p) and (it+p) data independently which allowed the analysis to go off-course. This was basically due to the poor quality of the (u~p) data which were used to obtain the T = 3/2 amplitude without any help from (n +p) data for the determination of this amplitude. A recent publication by Frank et a l . (FRA83) from LAMPF has also caused + some confusion. They have investigated (it p) elastic scattering in the energy range of 21 to 96 MeV and their results are often in disagreement with the KH and VPI phase shift analyses which, after a l l , are only a summary of previous measurements. The main problem is with the overall normalization, as the shape of the differential cross section seems reasonable. The disagreement with the Zid e l l analysis is not important, as this analysis proved incorrect at low energies and has since been revised. Of particular concern is the normalization discrepancy of the LAMPF data at 96 MeV, because this is close to precise and consistent total cross section measurements from CERN and SIN (PED78). There are also other LAMPF data of Ritchie et a l . (RIT83). The analysis of another u _p elastic scattering experiment by the TRIUMF-Colorado group at T^ - 70, 90, 120, 130, and 143 MeV is in progress. The preliminary results of this experiment are consistent with the KH analysis and do not support the LAMPF results. As for the scattering length, our results are in good agreement with Duclos et a l . (DUC73), who obtained an estimate of ^ - 8 3 ) from their 180° pion charge exchange cross section measurements at T =22.6, 32.9, and - 170 -42.6 MeV. The charge exchange values for (a 1-a 3) can also be compared with those obtained from (rt^p) elastic scattering (BUG73, W0074), which under the assumption of charge symmetry should be the same. There are also other values available which are normally the result of more indirect approaches. The most common one is from phase shift analyses of a l l the lower energy pion-nucleon data. Table V .5 presents the result of several recent (up) amplitude analyses and their scattering lengths. Apart from the anomalously high value of Z i d e l l et a l . , these values are in agreement with each other and also with our result. As most of these results are effectively analyses of (u ±p) elastic scattering data, we can see that there is support for charge independence, but because of the inconsistencies among the analyses of the same scattering amplitudes, i t is d i f f i c u l t to give a very good limit on any breakdown of isospin invariance. However, comparing these numbers to our own experimental result, we conclude that there is no evidence of violation of charge independence in the (TCN) system to a level of 4% in the amplitudes. An alternative approach is to use the Brueckner, Serber, and Watson relation to obtain the scattering length from the photoproduction reaction (yp •*• u +n) . Such an analysis by Spuller et a l . (SPU77) gives excellent agreement with our direct results . They deduced the (a^-a 3) scattering length from photo-production data using their measurement of the Panofsky Ratio. Another technique, though less accurate, is the determination of the 2p-ls transition energy and thus the strong interaction shift in pionic hydrogen (F0S83, B0V85). Pionic hydrogen Is an elementary atomic system composed of a bound proton and a u -, and the strong Interaction manifests - 171 -i t s e l f mainly while the pion resides in the atomic Is level before being absorbed by the nucleus, and produces a shift in energy as well as an increase in the width of the 2p-ls atomic transition. In the absence of the strong interaction, the 2p-ls transition energy can be calculated to a sufficiently high precision as the sum of the energy eigenvalue of the Klein-Gordon Hamiltonian of a particle in a Coulomb f i e l d and a term to account for vacuum polarization. The strong interaction shift is the difference between this calculated value and the experimentally measured energy. From an experimental point of view, i t is d i f f i c u l t to determine the strong interaction shift accurately because of the low event rate, and the small size of the energy s h i f t . The strong Interaction shift, c, is related to the s-wave low-energy pion-proton scattering length, a(u~p), by the approximate relation (DES54, BRU55) where E. is the binding energy of the it~ In the Is orbit and r is the corresponding Bohr radius. This scattering length may be expressed (DES54) In terms of the isospin components a^ and a 3 , a(n~p) = 1/3 {la^+a^t * n order to allow a comparison with low energy scattering experiments. The experimental situation can be compared with the theoretical expectations for explicit charge symmetry breaking calculations within the bag model. These arise basically due to the difference of the Chiral coupling of external pions to interior up and down quarks, because of their mass - 172 -difference. For example, the recent calculations of Thomas et a l . (TH081) and Pascal et a l . (PAS82) predict a difference of the (ppu°) and (nnrc0) coupling constants of less than 0.5%. The experimental sensitivity i s presently inadequate by approximately an order of magnitude to detect such effects . It is interesting that there is a similar situation around T 500 to 600 MeV. TC + Recent (TC p) elastic data (G0R81, SAD82, ABA83) are s t i l l inconsistent with charge exchange cross sections at backward angles, especially around 180°, and i t seems likely that, as at our energies, the early measurements of the charge exchange cross section are inaccurate. V.5.3 Pion-Nucleus Interaction and Free Pion-Nucleon Excitation Function At energies below 100 MeV, the interaction of pions with nuclei is funda-mentally different from the interaction near the energy of the (3,3) resonance. The pion-nucleus total cross section is much smaller and the nucleus appears relatively transparent in comparison to i t s strongly absorptive nature near the it-nucleon (3,3) resonance region. Also, at energies well below the (3,3) resonance the forward angle u-nucleon charge exchange is characterized by an interference phenomenon with a sharp energy dependence. This is caused by a nearly complete cancellation between the repulsive s-wave and attractive p-wave isovector interactions, and results in a very deep and sharp minimum in the forward angle Tt-nucleon charge exchange cross section near 40 MeV. The appearance of this cancellation could be substantially modified in nuclei because of medium effects. Its manifestation and existence in nuclei w i l l depend on the relative contribution of the s- and - 173 -p-vave parts of the amplitude. Because these contributions can be altered by effects such as nucleon correlations, Paul! blocking, and true absorption, the investigation of forward angle pion single charge exchange (SCX) to the isobaric analog state (IAS) in this energy region can be a sensitive method for the study of nuclear medium effects, and a cornerstone for the construction of a more complete picture of low energy u-nucleus interactions, along with elastic scattering and double charge exchange (DCX) reactions. The latter involves two nucleons, and i t w i l l be necessary to understand the sequential (SCX) contribution to (DCX) before conclusions about other effects, such as correlations, can be found (MIL84) . The disappearance of the forward-angle u-nucleon charge exchange cross section carries over into the pion-nucleus S C X reactions. Thus i f DCX proceeds by two successive forward SCX scatterings, with the IAS as the dominant intermediate state, i t should be highly suppressed. In contrast with this reasonable expectation, the experiment indicates that the DCX angular distribution at about 40-50 MeV on  l h C and 1 8 0 are forward peaked and the cross sections are much higher than expected (LEI85, ALT85) from conventional models. This observation of large, forward peaked cross sections in low energy pion-nucleus DCX reactions could provide evidence that six-quark, hidden color components of nuclear wave functions exist (MIL84). Fig. V.17 presents the angular distribution for 1 5 N ( T C + , T C ° ) 0 1 5 ( I A S ) at several energies. It is found that for pion (SCX) to the (IAS), the shape of the angular distribution Is very energy dependent, reflecting the analogous - 174 -0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 0 c m ( d e g ) Figure V.17 Angular distribution IAS at different energies. The curves are the result of a Legendre polynomial f i t to the data and are used to extrapolate to the cross section at zero degree (IR085). - 175 -behaviour of the free nucleon (SCX) cross section (IR085). It has a f a i r l y f l a t shape at 32.4 MeV and a deep minimum at 0° for 48.2 MeV. As the pion energy is increased the minimum f i l l s in and at 63.6 MeV the angular distribution f i n a l l y becomes strongly forward peaked. This energy dependence is caused by a change in the relative strength of the s- and p-wave amplitude contributions to the interference. F i g . V.18 shows the measured (IR085) 0° excitation function for the (IAS) transition on 7 L i , 1 4C, 1 5N, 3 9K, and 1 2 0 S n . Except for 7 L i , which seems to violate the general systematics, a general n° picture emerges: (1) the location of the minimum, T , is constant within the m uncertainties; (2) the width does not increase substantially except for the heaviest nucleus, 1 2 0Sn; and (3) the cross sections per excess neutron at the minimum, a /(N-Z), are a l l s t a t i s t i c a l l y consistent with that of the free m nucleon charge exchange, although the errors are large. These features make the forward angle charge exchange (IAS) transition in nuclei appear very similar to that of the free nucleon charge exchange, which is surprising because nuclear medium effects might be expected to diffuse this phenomenon. A l l in a l l , this similarity indicates that the energy dependence of the isovector pion-nucleon interaction is not strongly modified by the nuclear medium. There are discrepancies between different phase-shift predictions (ROW78, ARN85, K0C80) for the position and depth of the minimum in the free it-nucleon excitation function. These phase shifts predict this minimum to be between 45 and 50 MeV. It is interesting to note that the location of the minimum in the 7 L I data (IR085) is lower by 3 to 8 MeV compared to the - 176 -TV ( M e V ) Figure ¥.18 The measured 0° excitation function for the free («N) process and the IAS transition on several nuclei (IR085). - 177 -Tf-nucleon minimum as predicted by various phase shifts. To resolve these discrepancies, and to perform a quantitative comparison of theory to experiment, would require a precise measurement of (it~p •*• Tf°n) at 0°. The result of our measurement at 0° and 180° i s presented in Table V.6 and Fig. V.19. At 0° our data are consistent with Fitzgerald's result (FIT85), however the old SAID phase shift analyses (FP85, SP85) do not f i t the experimental data but require a minor modification to the phase shifts. The most recent analysis, (SM86), f i t s best and is reproduced in Fig. V.19. The Karlsruhe phase shifts also f i t the 0° data reasonably well but not perfectly. (We note that the minimum that we obtain from the Karlsruhe analysis is 1.527 ub/sr at 45.9 MeV which is slightly higher than the line drawn in Fig. 4 of Fitzgerald et a l ; the rest of the curve we confirm. For (SM86) the minimum is 1.524 Ub/sr at 45.5 MeV.) As for 180°, the older SAID analysis, (FP85), f i t s the data quite well, but the most recent analysis, (SM86), and Karlsruhe analysis are both slightly higher than our results and those of Duclos et a l . It should be mentioned that, contrary to several comments in the literature, Duclos' results are not direct measurements of (a^-ag) and when their cross-sections are analyzed the use of different p-wave phase shifts results in f a i r l y different s-wave scattering lengths. V.5.4 Conclusion The present results have underlined the u t i l i t y of good y-ray detectors and enabled us to obtain accurate cross section for the ( T T - P •»• yn) and (Tf-p •*• Tf°n) reactions, which guarantee an improvement in our knowledge of the - 178 -TABLE V . 6 Differential cross sections at 0° and 180° for the (ir~p •»• ir°n) reaction, obtained from our best fits. Tfl (MeV) da 7EF (0°) [mb/sr) ^ (180°) (mb/sr) 27.4 (SAL83) r 0.03 ± 0.03 0.87 + 0.04 39.3 l-0.02 + 0.06 1.22 ± 0.07 45.6 0.01 + 0.04 1.43 + 0.05 62.2 0.12 ± 0.05 1.91 + 0.05 76.4 0.21 ± 0.07 2.54 + 0.07 91.7 0.73 ± 0.14 3.46 + 0.15 106.8 1.46 + 0.16 4.44 + 0.16 121.9 2.27 + 0.21 5.70 + 0.22 - 179 -10000 0 1 1 1 »—I 1 i i i_ 0 20 4 0 60 60 100 120 T,. (MeV) Figure V.19 The differential cross sections at 0* and 180° for (w~p + n°n) reaction Including Saclay data (DUC73) and recent Los Alamos data (FIT85). - 180 -low energy pion-nucleon Interaction. The radiative capture measurements f i t quite well into the existing picture and are consistent with earlier work, although much more accurate. They confirm the principle of detailed balance but indicate that there Is a clear difference between data and calculations for the relative weighting of multipole amplitudes, and the calculations do not reproduce the nucleon data adequately. Together with polarization data, they should also permit a multipole analysis to be completed for photomeson production on neutrons without having to rely on theoretical input. The results from the charge exchange cross sections at lower energies are substantially lower than previous direct measurements. At the higher energies, the charge exchange data are in excellent agreement with the total cross sections obtained by Bugg et a l . This provides an additional confidence in our radiative capture data which are obtained simultaneously. There is also excellent agreement between our results and the KH phase shift analysis as well as the latest VPI analysis by Arndt et a l . It should be mentioned that the charge exchange results from the f i r s t phase of this project, i.e. at T =27.4 and 39.3 MeV, are included in VPI data base and are the only low TT energy data therein, because a l l previous measurements were probably in error. The agreement with Rowe et a l . is adequate, whereas the disagreement with the published phase shifts of Zidell et a l . illustrates the remarkable discrepancy between the older data and the present results at lower energies. This is re-emphasized in Table V.5 where a selection of values for the SCX s-wave scattering length is li s t e d . The data are f i n a l l y settling down to a reasonable number of about 0.270 in natural units. It should also be + reiterated that there is s t i l l a lack of data on (TT p) elastic scattering at low energies, and so the phase shift analyses are not strongly constrained. - 181 -It would be important to check charge independence by a comparison between the elastic and charge exchange reactions. It used to be assumed that isospin invariance was a fundamental symmetry of the strong interactions. However, i t is becoming clear that i t i s most likely an effect caused by the fact that the constituent masses of the up and down quarks happen to be similar. If the concept of isospin i s no longer a basic symmetry, but an accidental one, i t becomes essential to determine i t s limitations and the low energy Tf-nucleon interaction is a good testing ground. There are already several indications that breakdowns might have already been observed, especially i n baryon masses and in the nucleon-nucleon interaction (ISG80). A detailed discussion of possible experimental tests is given in the proceedings of a workshop on charge symmetry held at TRIUMF (TRI81). The clearest evidence seems to be the observation of isospin forbidden decay •*• T T ° i p ) which is found to have a branching ratio of (0.09 ± 0.02 ± 0.01)% (0RE80, LAN80). 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(A-l) -1 < n < + 1 with the recurrence relation: (£+1) Pf+10x) = (2E+1V Pf0i) - f . (A-2) They satisfy the orthonomality conditions: J P£0)PrO)d(i= ^ %- (A-3) and special values are: P 0 O ) = l , i»0O = «. P 2 0 ) = | ( 3 ^ 2 - l ) . (A-4) - 193 -A N G U L A R MOMENTUM The associated Lengendre functions Pfa) are defined by: if to = (1 V)m/2 ^ fyto, m = 0, 1, 2, . . . £ (A-5) with: if to = P^) (A-6) and the recurrence relation: (f-1-m) P£™ ( » = (2£+l)^ If ^) - (f+m) . (A-7) They satisfy the orthonomality relations: JT If to Pf,to = 8ff« • (A-8) It is on occasion useful to employ the derivatives of the associated Legen-dre function, ^ [lfto]> which satisfy the recurrence relation: (1-fx2) ^ [Pftol = <MV f fy> " P&W • (A -9) The spherical harmonics m(^»<^) ate defined by: 1_ rt,uP<» = [%r ^S] 2 P™(cos 0) e™* £ = °'11 2 •• m= 0, 1, 2. ... £ and: (A-10) They satisfy the orthonomality conditions: f f Yl'ta'<04) Yi d<f> d(cos0) = S £ £ - 5 W • (A-12) - 194 -A.2. T H E CLEBSCH-GORDAN COEFFICIENTS T h e s p h e r i c a l harmonics are e igenfunct ions of L and L z > where L i s the orb i ta l angular momentum operator, be longing to the e igenva lues £(£+1) and m (where the un i ts are s u c h that h - 1). The phase convent ion i s observed that: L±Yttn(e,<f>),(Lx±iLy)Yttje,<f>) 1_ = + t£(£+l)-m(m±l)] 2 Ytiatl{64>) . (A-13) The Legendre po lynomia ls obey the f o l l o w i n g addi t ion theorem. If ( d j , ^ ) and (0 2,<£ 2) def ine two d i rect ions in s p a c e , such that 61+62<n and y i s the angle between these d i r e c t i o n s , then: c o s y = c o s f l j c o s d 2 + s i n f l j s i n # 2 cos(<£j — <£2) . The addi t ion theorem then reads that: £ A.2. THE CLEBSCH-GORDON COEFFICIENTS The C l e b s c h - G o r d o n c o e f f i c i e n t s are def ined as the transformation matrix: < JA, JB, mA, mB | ;', jA, jB, m > where JA, Jg are two commuting angular momentum vectors and J = JA + iB, and the e igenva lues of J 2 , J A , J ^ 2 ; J z , , JB are ;'0'+l)i )AtiA+l), iB(jB+l)i m. o>A, n>B r espect i ve l y . We employ the notation that: Cj j 0.»»4> DJb) s < jA, jB; mA, mB \ j; jA, JB, n? > . (A-15) - 195 -A N G U L A R MOMENTUM T h e s i m u l t a n e o u s e i g e n f u n c t i o n s o f J 2, J^ 2, }Q a n d J z a r e t h e n g i v e n i n t e r m s o f t he s i m u l t a n e o u s e i g e n f u n c t i o n s o f J^ 2, Jg2, a n d Jg by: \i> JA' JB' m > = S CUJB<J' m ; mfi)' 'A' 'B' Wa' W b > (A"16) mA,mB w i t h t h e i n v e r s e : \JA' >B' mA'mB>= X 2 S /^fi0' m; n,^> "fi51* m > • >=\>A % l m ( A - 1 7 ) T h e c o e f f i c i e n t C ( ; , m; n>j, m 2 ) v a n i s h e s u n l e s s : (a ) m = ir i j+ n? 2 (b ) -j < m < + j; -jl < ml < + >j; - > 2 < m 2 < + ; 2 ( A - 1 8 ) (O l;']-^! < i < (>i+;2) • T h e b a s i c s y m m e t r y r e l a t i o n s a r e : c j l j 2 ( - i ' m - mi'm2^ = Chh^'~m' - m i ' - m 2 ^ = ( - l / 1 7 2 ^ C • 0', m; m 2 , m . ) m l j2BiCJifi2,-m2,mv-m). (A-19) F o r i n t e g e r £, £ } , £ 2 , C £ p (f, 0 0 0 ) v a n i s h e s u n l e s s (£j + £ 2 +£) i s e v e n . W h e n o n e o f the a n g u l a r m o m e n t a v a n i s h e s , s a y ; 2 , w e h a v e : C > 1 > » ; » , . 0 ) = 5 j 7 i 6 • ( A - 2 0 ) - 196 -A.2. THE CLEBSCH-GORDAN COEFFICIENTS Tables of Clebsch-Gordon coeficients are given in Condon and Shortley (1935) for < 2 and by Simon (1954)* for ; < | . Tables A-l and A-2 give the values of the coeficients for the common cases in which one of the angular momenta has the value j or 1. Table A-l C ^ (j,w; m—n^,^) 'l2 m2 = \ 1 l 2 2 L2;,+1 _ 1 _2;,+ l J 3 2 >i+n,+ j ' 2jl+l 2>2+l «n2 = 1 Table A-2 Cj j O , m; m—m2,ir2) 0 -1 1_ l [<jl+m)(jl+«i+l'pp f"oi-m+l)(;1+ir,+ l)'j2 [{j1-m)(j1-m+1)1 L(2>1+ 1)(2;1+ 2) J [ (2;i+D0i+l) J L(2>i+1)(2>i+2)J h L 2 ; i 0 1 + D J U01+1)J L 2>l0l-l) J A - 1 E0'1-'") O'x-ni-t-1)|2 |(;1-ni)(;i+ni)]2 |0'1+'n+l)0'i + «n)| 2j101+D J |_ >,(2;1+ ) J |_ 2>j(2;1+l) J E . U. Condon and G. H. Shortley, The Theory of Atomic Spectre (Cambridge Univ. Press, Cambridge, 1935). A. Simon, Oak Ridge Nat. Lab. Report 1718 (1954). - 197 -A N G U L A R MOMENTUM A 3 . THE ROTATION MATRICES The rotation matrix 3V , (a,B,y) for a rotation defined by Euler angles m m (a, B,y) is defined as: P, (a,B,y) = <j,m'\ e~'aJ* e"^ e~'Y3*\j. m> . (A-21) Tn m It may be writen in reduced form as: 2)> , (a, B,y) = e+'(m'a~my) d> . (/3) (A-22) m m mm where: -i/3J ~ 01 01 (-1)* / o\2;'+m-m'-2A / o\m-m+2A 2* 0'-n'-A)!(>+ni-A)!(A+in-m)!A!\COS2J \ Si"2/ A where the sum is over al integer values of A for which the arguments of the factorials are non-negative. The symmetry properties of the dJ , (B) are: tn m dL ' m ^ > = < - 1 ) m d L - 0 5 > = d ' m „ - *L '(-^ > (A-23> m m nun —m,—m mm and: dL-JB) = (-1)ha'dL Jn-&=i-1^™ dL- • mm m ,—m m ,—m If the rotation (a,B,y) is the result of rotating first through (pl,Bl,y1) and then through (aj./Sj^L we have: dL'J^ = 2 d ™ ' M ' ^ 2 > d i-vW • (A-24) m m mm * mm A Special values for £ integer are: - 1 9 8 -A.3. THE ROTATION MATRICES rfJ0(/8)«Jfc(cos/3) = (-D"" [ S f f g l ] 2 f W / 8 ) (A-25) and. ^ > = # T ) ^ H ^ ^ - W - ) ^ ! ) ] 2 " ^ 005) (A-26) for ;' a half integer, with y = £ + L, special values are. £ fLL° 3 ) = (F7T) c o s (f) (*K-i ( c 0 BP> " p£'< c o s/ 3» 2 2 di i°S) = (PTT) s i n ( f ) (P£+i(cos # + p£'<cos 0» (A-27) 2 2 2 2 L -I di.if}=m c o s ( f ) [S p^cos & + y ¥ p £ ' ( c o s # ] The rotation matrices have the folowing symmetry properties: KmW,y) = (-Dm $Lm /S, y) ^ L ' ^ ' - f r - a ) • (A-28) They satisfy the Clebsch-Gordon series (A-29) - 1 9 9 -ANGULAR MOMENTUM with the inverse: H'm\ ^mia,B,y) (A-30) where in both eqs. (A-29) and (A-30) al the arguments of the rotation ma-trices are the same. The orthonomality properties of the rotation matrices are: m 2 sL1(a^'y)$L2(a^.y) = s n i i n i 2 • (A-32) m J / » 2 J T / V T « 2 7 7 c/n = j tfa dcosjS <fy, then: 'o 0 0 f<m ^ , B , Y ) ^ ( a ^ y ) = J*L 8^. 8 ^ 8 • ^ . (A-33) Special values of ' are: ^ . ^ y ) - ^ ^ ' " ) w i t h ' =0.1,2... - 200 -APPENDIX B TABLE B . l Variarions of parameters AQ , Aj and Aj with angle at fixed energy (T^ = 91.7 MeV ) for the charge exchange reaction. Since the errors assigned to these parameters from the f i t t i n g program were unreasonably small, we have chosen the standard deviations of these parameters for s t a t i s t i c a l errors. Angle AO A l *2 45° 1.27 + 0.01 -1.43 + 0.05 0.54 ± 0.30 60° 1.22 + 0.01 -1.38 + 0.08 0.92 ± 0.07 75° 1.22 + 0.02 -1.51 + 0.34 0.79 ± 0.06 90° 1.27 + 0.02 -0.98 + 0.20 0.88 ± 0.05 105° 1.39 + 0.03 -1.19 + 0.09 1.03 ± 0.11 120° 1.28 + 0.02 -1.44 + 0.10 0.30 ± 0.67 142° 1.46 + 0.03 -1.28 + 0.08 0.81 ± 0.08 Average 1.26 + 0.03 -1.37 + 0.10 0.83 ± 0.10 Figure B.l The variation of Legendre polynomial coefficient Aj with energy for the charge exchange reaction. Figure B.2 The variation of Legendre polynomial coefficient with energy for the charge exchange reaction. 

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